Exam 7: Discrete Probability

Find p(the number on the ball drawn is even).

Urn 1 contains 2 blue tokens and 8 red tokens; urn 2 contains 12 blue tokens and 3 red tokens. You pick an urn at random and draw out a token at random from that urn. Given that the token is blue, what is the probability that the token came from urn 1?

Prove or disprove: $p ( E \cup F ) = p ( E ) + p ( F ) \text { for all events } E \text { and } F$

In questions suppose you have a class with 30 students $\underline{\quad\quad}$ 10 freshmen, 12 sophomores, and 8 juniors. -You pick two students at random, one at a time. What is the probability that both are freshmen?

Find p(the number on the ball drawn is >3 ).

Let A be the set of all strings of decimal digits of length 5. For example 00312 and 19483 are two strings in A. You pick a string from A at random. What is the probability that (a) the string begins with 7575. (b) the string has no 4 in it. In questions 13-16 you have 40 different books (20 math books, 15 history books, and 5 geography books).

an experiment consists of picking at random a bit string of length five. Consider the following events: E1: the bit string chosen begins with 1; E2: the bit string chosen ends with 1; E3: the bit string chosen has exactly three 1's. - $\text { Find } p \left( E _ { 2 } \mid E _ { 3 } \right)$

Suppose you and a friend each choose at random an integer between 1 and 8, inclusive. For example, some possibilities are (3, 7), (7, 3), (4, 4), (8, 1), where your number is written first and your friend's number second. Find the following probabilities: (a) $p ($ you pick 5 and your friend picks $8 )$ . (b) $p$ (sum of the two numbers picked is $< 4$ ). (c) $p$ (both numbers match). (d) $p$ (the sum of the two numbers is a prime). (e) $p$ (your number is greater than your friend's number).

A pair of dice, each with the numbers 1, 2, 2, 3, 3, 3 on its six sides are rolled. (a) What is the expected value of the sum of the numbers showing? (b) What is the expected value of the product of the numbers showing in part (a)?

A die has the numbers 1, 2, 2, 3, 3, 3 on its six sides. If the die is rolled, what is the expected value and variance of the number showing?

A red and a green die are rolled. What is the probability of getting a sum of six, given that the number on the red die is even.

you flip a biased coin, where p(heads) = 3/4 and p(tails) = 1/4, ten times. -Find $p \text { (exactly } 9 \text { heads). }$

You pick two books at random, one at a time. What is the probability that the two books are from different disciplines?