3.2 Classical probability practice 3

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

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) 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).

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?