Regression to the mean (please help)

Question

Suppose Mike is playing a coin game as follows. First he throws three fair coins into the air (First Toss). For each “heads”, he wins a dollar. Then, he tosses the three coins again (Second Toss). For each “heads” on this second toss, he also wins a dollar. (So he can win anywhere from $0 to $6 in one play of the game.)

a)What is the probability of tossing three heads on the first toss?

b)What is the probability of tossing fewer than 3 heads on the first toss?

c)What is the probability of tossing 0 heads on the first toss?

d)Suppose Mike tosses 0 heads on the first toss. What is the probability of doing better than that on the second toss?

e)Suppose Mike tosses 0 heads on the first toss and then curses the coins (“You stupid coins!”) What is the probability of doing better on the second toss?

f)Suppose Mike tosses 3 heads on the first toss. What is the probability of doing worse than that on the second toss?

g)Suppose Mike tosses 3 heads on the first toss, and then praises the coins (“Good coins!”) What is the probability of doing worse on the second toss?

Answer 1

a) Each coin is independent and has an equal chance of heads and tails (fair coin). The probability of 3 heads is equal to (1/2)^3 or 1/8.

b) The probability of the coin tossing three heads on the first toss is the exact opposite of the first scenario. Either 3 heads will be tossed or fewer than three heads will be tossed and then encompasses the entire sample space. The probability of fewer than three heads is 1 minus 1/8 or 7/8.

c) Probability of 0 heads is the same as the probability of three tails. The probability of 3 tails is equal to (1/2)^3 or 1/8.

d) The first toss does not affect the second toss. The probability of getting more than 0 heads on the second toss is the same as saying 1 minus the probability of 0 heads or 1 - (1/2)^3. The probability of getting more than 0 heads on the second toss is 7/8.

e) Cursing the coins doesn't affect the outcome. Therefore this answer is the same as part d, 7/8.

f) The first toss does not affect the second toss. The probability of getting less than 3 heads on the second toss is the same as saying 1 minus the probability of 3 heads or 1 - (1/2)^3. The probability of getting less than 3 heads on the second toss is 7/8.

g) Praising the coins doesn't affect the outcome. Therefore this answer is the same as part f, 7/8.