Exam 7: A: Discrete Probability

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.

Urn 1 contains 2 blue tokens and 8 red tokens; urn 2 contains 12 blue tokens and 3 red tokens. You roll a die to determine which urn to choose: if you roll a 1 or 2 you choose urn 1; if you roll a 3, 4, 5, or 6 you choose urn 2. Once the urn is chosen, you 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?

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. -Find $p \left( E _ { 2 } \mid E _ { 3 } \right)$ .

What is the probability that a fair coin lands Heads 4 times out of 5 flips?

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. -Determine whether E2 and E3 are independent.

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. -Find $p \left( E _ { 3 } \mid E _ { 1 } \cap E _ { 2 } \right)$ .

You have two decks of 26 cards. Each card in each of the two decks has a different letter of the alphabet on it. You pick at random one card from each of the two decks. A vowel is worth 3 points and a consonant is worth 0 points. Let X = the sum of the values of the two cards picked. Find E(X), V (X), and the standard deviation of X .

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. -Determine whether $E _ { 1 } \text { and } E _ { 2 }$ are independent.

you pick a bit string from the set of all bit strings of length ten. -What is the probability that the bit string begins and ends with 0?

What is the probability that a randomly selected day of the year (366 days) is in May?

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.

a bowl has eight ping pong balls numbered 1, 2, 2, 3, 4, 5, 5, 5. You pick a ball at random. -Find p(the number on the ball drawn is $\geq 3$ ).

a bowl has eight ping pong balls numbered 1, 2, 2, 3, 4, 5, 5, 5. You pick a ball at random. -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?

you pick a bit string from the set of all bit strings of length ten. -What is the probability that the bit string begins with 111?

suppose you have a class with 30 students-10 freshmen, 12 sophomores, and 8 juniors. -You pick one student at random. What is the probability that the student is not a junior?

you pick a bit string from the set of all bit strings of length ten. -What is the probability that the bit string has the sum of its digits equal to seven?

In a certain lottery game, you choose a set of four different integers between 1 and 50, inclusive, and a fifth integer between 1 and 20, inclusive, which may be the same as one of the other four. (a) What is the probability you win the jackpot by matching all five numbers drawn? (b) What is the probability that you match three of the first four numbers, but not the fifth?

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)?

you flip a biased coin, where p(heads) = 3/4 and p(tails) = 1/4, ten times. -Find p(at least 7 heads).