553.421 Honors Intro. to Probability
Assignment #3
Due Friday, Sep. 23 11:59PM as a PDF upload to Gradescope.
3.1. You pick two cards from a standard deck of 52 cards. Compute the probability they are both red given they are the same color.
3.2. Consider the sample space of the 10! (equally likely) orderings of the digits 0 thru 9. Let’s say you’ll win if you pick an ordering that starts with three even digits or if you pick an ordering that ends with three odd digits. Compute the probability you’ll win. Please simplify completely!
3.3. (a) Suppose A and B are events. We denote set subtraction A - B to mean take away all the members of B that happen to belong to A. Since we can’t take away members of B that don’t belong to A, we can write A-B = A-(A∩B),i.e., A-B = A∩Bc. Show that P (A-B) = P (A)-P (A∩B). (b) We deal a 5-card hand from a standard deck of 52 cards. Let B be the event that all cards are black, let S be the event that all cards are the same suit.
(1) Is S ≤ B? Briefiy explain why or why not.
(2) Briefly, using words, describe the event B - S in the context of this problem. Then compute its probability.
3.4. A manufacturer is trying to promote a brand of cereal. They create a set of m = 4 action figures and, for each box of cereal, they choose one of these action figures equally likely at random to put in the cereal box (exactly one action figure per box). If a parent buys n = 8 boxes of this cereal, compute the probability they have collected all 4 action figures. Hint: If we let Fi represent the event that you’ve collected action figure i (i = 1, 2, 3, 4), then F1 ∩ F2 ∩ F3 ∩ F4 is the event that you’ve collected all four. Use the fact that (F1 ∩ F2 ∩ F3 ∩ F4 )c = F1(c) U F2(c) U F3(c) ∩ F4(c) .
3.5. (a) A and B are mutually exclusive and P (A) = .2 and P (B) = .3. Compute P (AjA U B).
(b) Re-do part (a) except now, A and B are not mutually exclusive but are independent, i.e., they have the property P (A ∩ B) = P (A)P (B).
3.6. Suppose we are told that 20% of college freshmen get homesick and 30% of college freshmen attend school more than 500 miles from home, and of the college freshmen that attend school more than 500 miles from home, 40% get homesick.
(a) What percentage of freshmen are both homesick and attend school more than 500 miles from home?
(b) Of the freshmen that get homesick, what percentage attend school more than 500 miles from home? (c) If a freshman is not homesick, what is the chance they attend school more than 500 miles from home?
3.7. I deal you 2 cards from a standard deck of 52.
(a) Given they are both red, compute the probability they are different ranks.
(b) Given they are different ranks, what’s the probability they are the same color?
3.8. 12 postal workers (9 females, 3 males) have applied for a promotion. Only two people get pro- moted and all 12 are equally qualified. Management decides to equally likely at random select two people from the 12 to be promoted.
(a) If a female is promoted, compute the probability that a male is promoted. (b) If a male is promoted, compute the probability that a female is promoted.
3.9. A parent has 2 children. If we know (at least) one of them is a girl, what’s the probability the other is a girl?
3.10. A fair coin is tossed repeatedly. If the first head happened on an odd-numbered toss (i.e., toss #1,3,5,7,. . . etc.), what’s the chance the 2nd head also occurs on an odd-numbered toss?
Hint: From a reduced sample space point of view, once we’re told that the first head happened on an odd-numbered trial, the coin flipping can be though of starting anew except now the first trail after this first head will be an even-numbered trial.
3.11. A computer generates a 5-digit code where each entry of the code is equally likely to be one of the 10 digits 0 thru 9 repetition allowed. If the code generated is nondecreasing order, what’s the probability it is missing the digit 5?
3.12. The multiplicative rule for conditional probability generalizes to many events, for example,
P (A1 ∩ A2 ∩ A3 ) = P (A1 )P (A2 jA1 )P (A3 jA1 ∩ A2 )
and
P (A1 ∩ A2 ∩ A3 ∩ A4 ) = P (A1 )P (A2 jA1 )P (A3 jA1 ∩ A2 )P (A4 jA1 ∩ A2 ∩ A3 ), etc.
(a) Please verify these two rules that are written.
(b) There are three boxes of marbles - call them box 1, 2 and 3. Box 1 initially has 2 blue and 1 green; Box 2 has 1 blue and 1 green; Box 3 has 0 blue and 1 green. The experiment is as follows: draw a marble at random from box 1, then put that marble into box 2; then draw a marble at random from box 2, and put that marble into box 3. Finally, draw a marble at random from box 3. Use one of the rules above to compute the probability that you select a blue marble at every stage of the experiment.
h.13. Let A1 , A2, . . . , An be any events. Provide a proof by mathematical induction (on n) to show that
h.14. Recall that in the matching problem we considered all n! orderings of the integers 1 thru n to be equally likely, and we let Mi be the event that there is a match at i. We showed the proba-bility that there no matches (i.e., a derangement) is Compute the probability that there is exactly one match. For example, the probability the one match is in position 1 is P (M1 only) = P (M1 M2(c)M3(c) · · · Mn(c)).
h.15. 12 balls (5 red, 4 green, 3 yellow) are lined-up left to right. If no two green balls are adjacent, what’s the probability the no two yellows are adjacent?