Probability Practice Problems

Here are several intermediate-level problems in probability for practice. These are not an assignment, and in fact some go beyond what you’re required to know for the coures. If you can do them comfortably then you should have no difficulty with the quiz. Hints are provided, but don’t look at them unless you absolutely have to!

1. A fair coin is flipped twice.

   (a) If you are told that at least one of the flips came up heads, what is the probability that both are heads?

   (b) If you are told that the first coin came up heads, what is the probability that both are heads?

   Answers:  (a) 1/3  (b) 1/2

   Need a hint? Write out the sample space. How many outcomes match what you were told? How many within that subset involve two heads?

2. A lottery has a $6,000,000 grand prize with probability of winning 1 in 3,000,000. It also has a $10 consolation prize with probability of winning 1 in 1000. What is the fair price of your $5 lottery ticket?

   Answers:  $2.01

   Need a hint? The fair price of a game equals the mean value or expected value of the probability distribution.

3. Police plan to enforce speed limits during the morning rush hour on four different routes into the city. The traps on routes A, B, C, and D are operated 40%, 30%, 20%, and 30% of the time, respectively. Biff always speeds to work, and he has probability 0.2, 0.1, 0.5, and 0.2 of using those routes.
   (a) What’s the probability that he’ll get a ticket on any one morning?
   (b) What’s the probability he’ll go five mornings without a ticket?

   Answers:  (a) 0.27  (b) .2073, or about .21

   Need a hint? What sort of events are “Biff takes route A to work today” and “there’s a speed trap on route A today”? What’s the probability Biff will get a ticket on route A in any one day — i.e., that he will take route A and there’s a speed trap on route A? Answer that question for the other three routes. Now, what sort of events are “ticket on route A”, “ticket on route B”, and so on?

4. In an urn are 5 blue, 3 red, and 2 yellow balls. If you draw 3 balls, what’s the probability that less than 2 will be red if—
   (a) You draw with replacement?
   (b) You draw without replacement?

   Answers:  (a) 98/125, or .785  (b) 49/60, or .8167

   Need a hint? Does it really matter that there are three colors? How can you relabel them to simplify the problem? Then, “less than 2 red” means how many red? Write out the sequences that lead to that many red, and assign probabilities.

5. Buffy takes a multiple-choice exam of 10 questions. Each of them has five possible answers (four wrong, one right). If Buffy knows the right answer she picks it; otherwise she picks randomly. Suppose she knows the answer to 70% of the questions.

   (a) What is the probability that she answers any randomly selected question correctly? P(right) = ?

   (b) If she gets a particular question right, how likely is it that she knew the answer? P(knew | right) = ?

   (c) What’s her expected score on this exam?

   Answers:  (a) .76  (b) about .921  (c) 7.6 correct

   Need a hint? Make a two-way table, with “Knows” and “Doesn’t know” columns and “Correct” and “Wrong” rows. Add Total row and Total column. Fill in cells based on what you know, then add or subtract to fill in the rest.

6. Butch will miss an important TV program while taking his statistics exam, so he sets both his VCRs to record it. The first one records 70% of the time, and the second one records 60% of the time. What is the probability that he gets home after the exam and finds—
   (a) No copies of his program?
   (b) One copy of his program?
   (c) Two copies of his program?

   Answers:  (a) .12  (b) .46  (c) .42

   Need a hint? One and only one of parts (a), (b), and (c) must happen, and therefore the three probabilities must sum to what number? If you can solve any two of them, you can use that fact to solve the third.