3.3 Conditional probability practice 3

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 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”.