553.420/620 Probability
Assignment #04
1. We have two spinners: spinner 1 has 5 equally likely regions numbered 1 through 5, spinner 2 has 6 equally likely regions numbered 1 through 6. You spin each spinner once and note the region number the spinner landed.
(a) What’s the probability you landed on at least one even number?
(b) Given you landed on at least one even number, what’s the probability the other number is odd?
Hint: for (a) it might be helpful to define Ei to be the event that spinner i landed even. Then E1 ∪ E2 is the event you landed on at least one even.
2. (a) Flip a fair coin twice. One of the flips is a head. What’s the probability the other is a tail?
(b) The Smith family has two children. One of them is Tina – a girl. What’s the probability Tina has a brother?
3. (a) In dealing two cards, what’s the probability we get a heart followed by a red card?
(b) In dealing two cards, what’s the probability you get a red card followed by a heart?
(c)* In dealing five cards, what’s the probability we see a black followed by black followed by red followed by a spade followed by a heart?
* just as in part (b) if we invoke some nice property of the sample space this problem is not too bad.
4. (a) Roll a 6-sided die twice. Given a 1 occurred, what’s the probability the first one shows a 6?
(b) Roll a 6-sided die twice. Given exactly one 1 occured, that’s the probability the first is a 6?
5. Among two coins – one fair, the other two-headed – one is selected uniformly at random.
(a) The coin is flipped twice. Let Hi be the event that we get a head on the ith flip, i = 1, 2. Are H1 and H2 independent? Justify your assertion.
(b) Flip the coin n times, where n > 1 be an integer. If the selected coin came up heads on all n flips, what’s the probability the two-headed coin was selected?
6. I deal each of three people 3 cards. What’s the probability at least one person has all red cards?
7. We have two boxes each filled with 2 red and 1 black balls. Two balls are drawn uniformly at random from one other these boxes and put into the other and then two balls are drawn from this box. What’s the probability we draw 2 red balls?
8. Two dice are selected without replacement from a 4-sided and two 6-sided dice, and rolled. If the sum total on the dice is 5 compute the probability we selected both 6-sided dice.
9. Angela hits the bulls-eye of a dartboard with probability 5/1 independently from throw to throw.
(a) What’s the probability with three (3) throws, Angela hits the bulls-eye all three times?
(b) If Angela hits the bulls-eye exactly once in three throws, what’s the probability she hit the bulls-eye on her first throw? There’s a very intuitive answer and a more rigorous one, provide both.
(c) If Angela hits the bulls-eye at least once in three throws, what’s the probability she hit the bulls-eye on her first throw?
10. (continued from problem 9.) Brendan hits the bulls-eye of a dartboard with probability 4/1 independently from throw to throw.
(a) Angela and Brendan each throw one dart. If the bulls-eye is hit (by one or both of them) what’s the probability Angela hits the bulls-eye?
(b) Angela and Brendan each throw one dart. If the bulls-eye is hit (by one or both of them) what’s the probability Brendan hits the bulls-eye?
(c) Angela and Brendan each throw one dart. If the bulls-eye is hit (by one or both of them) what’s the probability they both hit the bulls-eye?