Exam 7: Discrete Probability

Find E(X), where X = the number on the ball you draw.

You pick a word at random from the set of all words of length six of letters of the alphabet with no repeated letters. What is the probability that the word has exactly one vowel?

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

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

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

you flip a biased coin, where p(heads) = 3/4 and p(tails) = 1/4, ten times. -Find $p \text { (exactly } 7 \text { heads). }$

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 E1 and E2 are independent.

What is the probability that a fair coin lands Heads 6 times in a row?

of your numbers match the six winning numbers. In questions 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?

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

Find and correct the error in the solution to the following problem: Problem: You flip two coins and want to find the probability that both coins show heads.

You pick one book at random. What is the probability that the book is a history book?

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 the second student is a freshman, given that the first is a freshman?

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

Three coins are tossed. (a) List the elements in the sample space. (b) Find the probability that exactly two heads show.

You pick a word at random from the set of all words of length six of letters of the alphabet with no repeated letters. What is the probability that the word begins and ends with a vowel?

What is the probability that the sum of the numbers on two dice is even when they are rolled?

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

What is the probability that a randomly selected integer chosen from the first 100 positive integers is odd?

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.