HPC Ch3 practice

A circle is circumscribed around a right triangle (the hypotenuse of the triangle contain the center of the circle). If Louis randomly chooses a point inside the circle, what is the probability that the point is inside the shaded region?

Given the probabilities [math]P(A)=3\cdot P(B)[/math], [math]P(C)=0.5[/math], [math]P(A\cap B)=0.18[/math], [math]P(B\cap C)=0.1[/math], [math]P(A\cap C)=0.1[/math], [math]P(A\cap B\cap C)=0.08[/math], and [math]P(A\cup B\cup C)=1[/math]. Find [math]P(A)[/math].

A bag contains 5 gold, 4 silver coins and a bronze coin. Three coins are drawn from the bag without replacement. Find the probability that

(a) All are the same

(b) All are different

(c) Exactly two of the three are the same

(Calculator) Which one of these three events have the highest probability of occurring?

A fire station gets an emergency call from a shopping mall in the mid-afternoon. From a study of traffic patterns, the chief knows that probability the most direct route is clogged with traffic is 0.07, while the probability of the secondary route being clogged is 0.05. The probability both are clogged is 0.02. What is the probability they can respond to the call unimpeded using one of these routes?

Four people each rolled two dice 5 times. The results are summarized below. Evaluate the following experimental probabilities

(a) [math]P([/math]Someone Rolls two prime numbers[math])[/math]
(b) [math]P([/math]Someone Rolls at least one 3[math])[/math]
(c) If they continued playing for 1000 rounds, what value you expect part (a) and (b) to approach?

(Calculator) A hospital, Mas has implemented a power backup system with three independent backup electric generators. In the event of an electrical outage, two of the three generators have to function to provide the hospital with sufficient electricity. Each independent electric generator has a 0.971 probability of functioning property. What is the probability that NOT enough electricity is supplied by the electric generators during an electricity outage?

A square is inscribed in a circle that is inscribed in a square. If Ethan randomly choose a point in large square, what is the probability that the point is in the shaded region?

(Calculator) Darren rolls a fair cube 10 times. What is the probability of the following outcomes?

(a) There are exactly 6 even and 4 odd rolls

(b) There are at least 3 rolls that are 6s

(c) Somewhere in the 10 rolls, there are exactly 6 consecutive 1s in a row?
(A) [math]\dfrac{1296}{6^{10}}[/math]
(B) [math]\dfrac{4860}{6^{10}}[/math]
(C) [math]\dfrac{6480}{6^{10}}[/math]
(D) [math]\dfrac{8640}{6^{10}}[/math]

(Calculator) Ms. Tarr has two bags of candy. Bag A contain a mixture of 40% gummy bears and 60% jelly beans. Bag B contain a mixture of 15% gummy bears and 85% jelly beans. In a separate bowl, Ms. Tarr created a mixture using 75% from bag A and 25% from bag B.

(a) If you select a piece of candy selected at random from the bowl, what is the probability that it is a jelly bean?

(b) If you select 10 pieces of candy at random from the bowl (with replacement), what is the probability that you selected at least 7 jelly beans and that the first piece you selected was a gummy bears?

(c) Ms. Tarr wants to create a new mixture that contains 32% gummy bears. What percent of each bag did Ms. Tarr use to create this mixture?

(Calculator) You have twenty ping pong balls labeled with the numbers one through twenty. You randomly select 5 from a bag without replacement.

(a) What is the probability that you select all even numbers?

(b) What is the probability that you select all prime numbers?

(c) What is the probability each ping pong ball that you select is progressively larger than the previous one? (i.e. 1, 3, 4, 6, 10).

(Calculator) A pizza cut into 8 slices. 10 slices of pepperoni are randomly placed on the pizza. When you randomly select a slice of pizza, what is the probability of taking a slice that contains no more than 2 slices of papperoni?

A bag contains 10 red marbles, 10 green marbles, and 6 white marbles. Two marbles are drawn at random without replacement.What is the probability that:

(a) They are both red?

(b) They are different colors?

(c) One of them is green?

(d) There is at least one red?

(Calculator) Students A, B, and C are on the basketball team. They are taking turns shooting a free throw. Student A can make the shot 80% of the times, student B can make the shot 70% of the time, and student C can make the shot 85% of the times. If they each attempt one free throw, what is the probability that at least one of them makes the shot?

(Calculator) The Powerball lottery in the USA recently had a grand prize of $1.6 billion. The game involves picking five white balls from a drum with 69 balls (numbered 1-69 and the order they are pulled from the drum does not matter) and one red ball out of a drum of 26 balls (number 1-26).

(a) How many combinations of 5 white numbers and one red number are there?

(b) You win the grand prize if you match all the white numbers and the red number. What is the probability that your lucky number is a big winner? Use a fraction to express your answer.

(c) The chance of being hit lightning in your lifetime is 1 in 3,000. What can [math]n[/math] be so that the chance of being hit by lightning n-times is more likely than Powerball lottery?

(d) You win second prize, $1,000,000, if you match the 5 white balls. What is the probability of winning second prize? Use a fraction to express your answer.

(e) It cost $2 to play this game. If the grand prize is $ 1.6 billion, what is the mathematical expectation (expacted value) of the Powerball lottery?

You have 10 balls in a bag: 1 blue, 3 yellow and 6 green. If you pick three balls out of the bag:

(a) What is the probability that all are green?

(b) What is the probability that at least one is yellow?

(c) What is the probability that all are different?

(Calculator) From out class of 14 students, 9 are boys and 5 are girls. If pick 5 people at random to gain an extra credit point, what is the probability that:

(a) All are boy?

(b) Exactly 2 are boys and 3 are girls?

(c) What is the most common combination of boys and girls?

25% of us (humans) are unable to taste the bitter flavor of common foods such as raw broccoli, cabbage, coffee and dark beers. If you pick 5 people at random, what is the probability that: (3 decimal places)

(a) None of them taste bitter foods?

(b) Exactly 1 of them tastes bitter foods?

(c) Exactly 2 of them tastes bitter foods?

(d) That at least one of them does not taste bitter foods?

Kelly conducted a poll of 175 supermarkets has discovered that 115 of them sell [math]H\ddot{a}agenDazs[/math] ice cream and 122 of them sells [math]M\ddot{o}venpick[/math] ice cream.

(a) How many supermarkets sell both if each supermarket sells at least one of the two brand of ice cream?

(b) What is the probability that a store sells both brands given that they sell [math]M\ddot{o}venpick[/math]?

If events [math]A[/math] and [math]B[/math] are given such that [math]P(A)=x[/math], [math]P(A\cup B)=y[/math] and [math]P(B)=z[/math], explain how you can mathematically prove the following:

(a) Event [math]A[/math] and [math]B[/math] are independent.

(b) Event [math]A[/math] and [math]B[/math] are mutually exclusive.

Given the two events [math]A, B[/math] such that [math]P(A|B)=0.2[/math] and [math]P(B|A)=0.7[/math]. If events [math]A[/math] and [math]B[/math] are independent, find

(a) [math]P(A\cap B)[/math]

(b) [math]P(B)[/math]

(c) [math]P(B’\cap A’)[/math]. ([math]A'[/math] is the complement of [math]A[/math])

(Calculator) Given that [math]P(C)=\dfrac{9}{20}, P(C| D’)=\dfrac{3}{7} and P(C| D)=\dfrac{6}{13}[/math]

(a) Find P(D)

(b) Find [math]P(C’ \cap D’)[/math]

(c) Are [math]C[/math] and [math]D[/math] independent? Explain.

In a group of 40 students, 22 are in Honors Algebra 2, 25 take AP World History and 3 take neither of these subjects. Determine the probability that a randomly chosen student is in:

(a) Both Honors Algebra 2 and AP World History.

(b) At least one of these subjects.

(c) Honors Algebra 2 given that he or she is in AP World History.

(Calculator) Every Monday there is a [math]\dfrac{2}{5}[/math] probability that Alex is sleep deprived in class. There is only a [math]\dfrac{1}{10}[/math] chance that he is sleep deprived in class on any other weekday.

(a) What is the probability that Alex is sleep deprived in class?

(b) If Alex is sleep deprived, what is the probability it is a Monday?

You flip a coin four times in a row and get heads each time.

(a) What is the empirical probability of getting heads in this experiment?

(b) How does this compare to the theoretical probability of getting heads on a fair coin?

You and a friend are playing Yahtzee. Yahtzee is a game where players roll 5 dice an try to collect sets of numbers. On each turn, player is given 3 rolls of the dice. The first roll is of is of all 5 dice together. The player then selects 0-5 dice to “re-roll”. She re-rolls these dice. The player again selects 0-5 dice to re-roll a seconds times. She re-rolls these dice then, selects how to score the dice. Answer the following questions in fraction. Note that [math]6^4=1296[/math]

(a) Yahtzee is 5 of the same number. What is the probability that you roll the dice and get a Yahtzee?

(b) A four-of -a-kind is four matching dice and one die that does not match. What is the probability of getting a four-of-a-kind on the first roll?

On your first roll you get: . You decide to go for a Yahtzee, a four-of a kind, or a full house (three of a kind and two of a kind). So you keep the and roll the again.
(c) On your second roll,what is the probability that you roll these two dice and get two 2s, or a 2 and some other number, or two of kind of some other number?

(d) Starting with ,what is the probability that you get Yhatzee in the next two rolls? That is ,find the probability of turning theinto within two rolls.

Box A contains a red marble, a blue marble, and a yellow marble. Box B contains a red marble, a blue marble, a yellow marble and a green marble. One marble is taken from each box. What is the probability that both marbles are the same colour?

200 people with neurology symptoms, which include headaches and backaches, participated in a study to evaluate a pain relief medicine. All the people took the medicine and the results were as follows:

• 60 people experienced headache relief
• 126 people experienced backache relief
• 6 people experienced relief from both

What is the probability that a person who takes the drug experiences relief from (use fractions to express your answers)

(a) At least one of the two symptoms?

(b) Neither of the symptoms?

Identify the following scenarios as independent or dependent events and then answer the question. If the scenarios is independent, type “A”. Otherwise, type “B”

(a) A wedding planner has 3 personalized invitations for Alice, Bob, and Carol that belong in 3 addressed envelopes (Alice’s envelop, Bob’s envelope, Carol’s envelope). If he randomly selects an envelope for each invitation, what is the probability that all 3 cards are placed correctly?

(b) What is the probability of rolling a pair of dice twice in a row and getting a sum of 7 both times?

(c) What is the probability of drawing a face card (J, Q, K) and then drawing an Ace from a complete 52 card desk that is well-shuffled?

Identify each as a mutually exclusive or inclusive event and then answer the question. If the scenarios is exclusive, type “A”. Otherwise, type “B”

(a) The probability of rolling a 1 or a 2 on a single die

In a class of 20, half of the students are girls and 15 of them area the oldest in their family. 4 of the girls are not the oldest in their families.

(b) The probability that a student is a girl or the oldest?

(c) The probability that a student is a boy or is the oldest?

A bag contains balls numbered 1 through 30. Two balls are chosen at random with replacement. What is the probability that: (use fractions to express your answers)

(a) both numbers are odd?

(b) one number is even and the other is odd?

(c) both numbers are less than 10.

(d) the same number is chosen twice?

Two six-sided dice are rolled. Find the probability that the sum is greater than 7 or the sum is even.

In an exam, two independent reasoning problems, 1 and 2, are asked. 35% students solved problem 1 and 15% students solved both the problems. What percentage of students who solved the first problem will also solve the second one?

In a high school graduating class of 128 students, 52 are on the honor roll. Of these, 48 area going on to college; and of the other 76 students, 56 are going to college. If a student is selected at random from the class, what is the probability that the person chosen is: (use fractions to express your answers)

(a) going to college?

(b) not going to college?

(c) not going to college and on the honor roll?

(Calculator) A shipment of 25 television sets contains three defective units. What is the probability that a vending company purchasing four of these units and receive

(a) All good units?

(b) Two good units?

(c) At least two good units?

(Calculator) Three prizes are rewarded in a raffle during a halftime show at a school basketball game. Ben, James, Ashley, and [math]17[/math] other students each have one ticket. Use fractions to express your answers

(a) If the raffle has a first, second, and third prize, what is the probability, as an exact value, that Ben wins first prize, James wins second prize, and Ashley wins third prize?

(b) If the raffle has three identical prizes, what is the probability that Ben, James, and Ashley win the prizes?

Chess is a game played by two players. A game results in one of three mutually exclusive outcomes-either player A wins, or player B wins, or the game results in a draw. In a world championship chess match, the Russian champion Kozlov and the American champion Armstrong are involved in a sudden death payoff. The first player to win a match wins the championship. From previous results, the probability that Kozlov wins a match is [math]\dfrac{1}{3}[/math] and the probability that Armstrong wins a match is [math]\dfrac{1}{4}[/math]. Calculate the probabilities that: (use fractions to express your answers)

(a) the first match is a draw.

(b) the championship has not been decided after the second match.

(c) Armstrong wins within two matches.

(Calculator) From [math]6[/math] biologists, [math]4[/math] chemists, and [math]3[/math] physicists, a scientific committee of [math]5[/math] is to be chosen. The top two scientists-April, a biologist, and Lee Kim, a chemist must be on the committee.

(a) What is the probability that at least [math]3[/math] biologists are chosen?

(b) What is the probability that at least [math]1[/math] physicist is chosen?

(c) What is the probability that the [math]3[/math] remaining members chosen are all from the same branch of science?

A golfer is practicing putting. The probability that he holes the first putt is [math]0.4[/math]. Each time he holes a putt the probability that he holes the next putt increases by [math]25\%[/math]. Each time he misses a putt the probability that he misses the next putt increases by [math]20\%[/math]. Calculate the probability that: (use fractions to express your answers)

(a) he holes the first two putts

(b) he misses the first two putts

(c) he holes two of the first three putts

A gambling casino has three gaming machines of two types, [math]X[/math] and [math]X'[/math], which look identical. The probability of choosing [math]X[/math] is [math]\dfrac{1}{3}[/math], and the probability of picking [math]X'[/math] is [math]\dfrac{2}{3}[/math]. (There are two [math]X'[/math] machines.) The probability of a win with machine [math]X[/math] is [math]\dfrac{1}{3}[/math] while the probability of a win with machine [math]X'[/math] is [math]\dfrac{1}{4}[/math]. Mr. A. chose a gaming machine at random and won. What is the probability Mr. A. chose machine [math]X[/math]?

(Calculator) Weather records indicated that the probability that a particular day is dry is [math]\dfrac{3}{10}[/math]. Liverton Villa is a football team whose record of success is batter on dry days than on wet days. The probability that Liverton win on dry day is [math]\dfrac{3}{8}[/math], whereas the probability that they win on a wet day is [math]\dfrac{3}{11}[/math]. Liverton are due to play their next match on Saturday.

(a) What is the probability that Liverton will win ?

(b) Three Saturday ago Liverton won their match. What is the probability that it was a dry day?

Given that [math]P(A)=0.75, P(B|A)=0.8[/math] and [math]P(B|A’)=0.6[/math] calculate [math]P(B)[/math] and [math]P(A|B)[/math].

Pablo drive to work. The probability that he leaves home before 7:00 is [math]\dfrac{3}{4}[/math]. If he leaves home before 7:00, the probability he will be late for work is [math]\dfrac{1}{8}[/math]. If he leaves home at 7:00 or later, the probability he will be late for work is [math]\dfrac{5}{8}[/math]. Use fractions to express your answers

(a) Find the probability that Pablo leaves home before 7:00 and is late for work.

(b) Find the probability that Pablo is late for work.

(c) Given that Pablo is late for work, find the probability that he left home before 7:00.

(d) Pablo will drive to work two days next week. Find the probability that he will be late at least once.

A large group of university students take three examinations in the course of a semester. It is found that if a student passes an examination, the probability of his passing the next one is four-fifths, and if he fails an examination, the probability of his failing the next one is also four-fifths. In the first examination, [math]60\%[/math] of the students pass.

(a) What is the probability that a student passes all three examinations?

(b) If a student passes exactly two examinations, what is the probability that he failed the first one?

Ethan has [math]18[/math] pens. [math]12[/math] are made of plastic and [math]13[/math] are red. [math]2[/math] are neither red nor plastic. Ethan chooses a pen at random.

(a) Find the probability that it is a red plastic pen.

(b) If it is a red pen, find the probability that it is plastic.

One card is drawn at random from a deck of [math]52[/math] cards. The following events are defined:
A: a diamond is drawn B: a ten is drawn C: a red card is drawn

(a) [math]P(A \cup C)[/math]

(b) [math]P(A \cap B’)[/math]

(c) [math]P(A|C)[/math]

(d) [math]P(C|A)[/math]

Given[math]P(T) = 0.34,P(A) = 0.28[/math], and [math]P(T \cup A)= 0.47[/math]

(a) Are events T and A are mutually exclusive events? Type “Yes” or “No”.

(b) Find [math]P(T\cap A)[/math].

(c) Are events T and A are independent? Type “Yes” or “No”.