Due: Monday, October 2, at the beginning of class.
Show your work for each problem. Don't guess. You should be able to convince a skeptical person that your answer is correct. Answers are in blue.
An urn contains 10 balls, numbered from 1 to 10.
If you select one ball at random from the urn, what is the probability that the ball is ball 3?
If you select two balls at random, without replacing the first one after the first selection, what is the probability that one of the balls is a 3?
If you select two balls at random, replacing the first one back into the urn after the first selection, what is the probability that at least one of the balls is a 3?
If you select two balls at random, replacing the first one back into the urn after the first selection, what is the probability that the two balls have the same number?
Suppose that three balls are chosen at random without replacement. What is the conditional probability that the third ball is ball 4, given that the first two balls are 1 and 5, in that order?
Suppose that three balls are chosen at random with replacement. What is the conditional probability that at least two of the three balls have the same number, given that the first two balls are 1 and 5, in that order?
Two of the balls will be the same provided the third ball is ball 1 or ball 5. There is a 2/10 = 1/5 chance of that happening.
Suppose that three balls are chosen at random with replacement. What is the conditional probability that all three balls have the same number, given that the first two balls have the same number?
A deck of cards consists of 52 cards. There are 13 different card ranks: ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen and king. There are 4 cards of each rank in the deck, one of each suit: spaces, clubs, hearts and diamonds.
A poker hand consists of 5 cards drawn from a deck of cards.
How many different poker hands are there?
If a hand is chosen at random (all hands equally likely) what is the probability that the hand contains an ace?
(48 choose 5)/(52 choose 5) = (48×47×46×45×44)/(52×51×50×49×48) = (47×46×45×44)/(52×51×50×49) = 4280760/6497400 ≈ 0.659.The probability of a hand containing an ace is about 1 − 0.659 = 0.341.
A flush is a hand where all of the cards have the same suit. What is the probability that a randomly chosen hand is a flush?
4(13 choose 5)/(52 choose 5) = (4×13×12×11×10×9×5×4×3×2)/(5×4×3×2×52×51×50×49×48) = (4×13×12×11×10×9)/(52×51×50×49×48) = (12×11×10×9)/(51×50×49×48) = (12×11×9)/(51×5×49×48) = (11×9)/(51×5×49×4) = (11×3)/(17×5×49×4) = 33/16660 ≈ 0.00198
A straight is a hand whose cards consist of a sequence of ranks, without gaps. For example, a hand containing a 4, 5, 6, 7 and 8 is a straight. The suits of the cards does not matter. What is the probability that a randomly chosen hand is a straight?
The probability of getting a straight is 36×44/(52 choose 5).
36×44/(52 choose 5) = (36×44×5×4×3×2)/(52×51×50×49×48) = (36×44×5×3×2)/(13×51×50×49×48) = (36×42×5×3×2)/(13×51×50×49×3) = (36×42×5×2)/(13×51×50×49) = (36×42×5)/(13×51×25×49) = (36×42)/(13×51×5×49) = 576/162435 ≈ 0.00355.
A full house is a hand consisting of 3 cards of one rank and 2 cards of another rank. For example, a hand that has 3 jacks and 2 7's is a full house. What is the probability that a random hand is a full house?
After choosing the 3 cards that match, there are 12 ranks to choose from for the other 2 cards. There are (4 choose 2) = 6 ways of choosing two cards of that rank. So there are 12×6 ways to select the 2 matching cards.
The total number of full house hands is 13×4×12×6. The probability of choosing a full house is (13×4×12×6)/(52 choose 5).
(13×4×12×6)/(52 choose 5) = (13×4×12×6×5×4×3×2)/(52×51×50×49×48) = (12×6×5×4×3×2)/(51×50×49×48) = (6×5×3×2)/(51×50×49) = (6×3)/(51×5×49) = (6)/(17×5×49) = 6/4165 ≈ 0.00144
What is the conditional probability that a randomly chosen hand contains at least two jacks, given that it contains at least one jack?
There are (4 choose 2)(48 choose 3) = 103776 hands with exactly two jacks, (4 choose 3)(48 choose 2) = 4512 hands with exactly three jacks and 48 hands with exactly 4 jacks.
There are 103776 + 4512 + 48 = 108336 hands with two or more jacks. There are 778320 + 108336 = 886656 hands with at least one jack. So the conditional probability that a hand at least two jacks, given that it has at least one jack, is 108336/886656 ≈ 0.122.
What is the conditional probability that a randomly chosen hand contains at least three jacks, given that it contains at least two jacks?
What is the conditional probability that a randomly chosen hand is a straight, given that it is a flush?
There are 9 straight-flush hands in a given suit, so there are a total of 36 straight-flush hands. The conditional probability of selecting a straight-flush, given that you have selected a flush, is 36/5148 = 1/143 ≈ 0.07.