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# Probability

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Help meee hard problems

Valeria has three coins, each of which has a probability of 1/2 of coming up heads. On her first turn, she flips all three coins. If all three come up heads, then she stops. Otherwise, for her second turn, she flips any coins that came up tails.  If there are still coins that come up tails, she flips any coins that came up tails, and so on.

(a) What is the probability that Valeria has two heads on her first turn?
(b) What is the probability that Valeria has three heads on her first turn?

(c) What is the probability that Valeria has two heads after her second turn?

(d) What is the probability that Valeria has one head after her second turn?

(e) What is the probability that Valeria has one head after her third turn?

(f) What is the probability that Valeria has three heads after her fourth turn?

Mar 13, 2023

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(a) The probability that Valeria has two heads on her first turn is the probability that one of the coins is tails and the other two are heads. There are 3 ways that this can happen: the first coin is tails and the other two are heads, the second coin is tails and the other two are heads, or the third coin is tails and the other two are heads. Each of these three outcomes has probability (1/2) * (1/2) * (1/2) = 1/8. Therefore, the probability that Valeria has two heads on her first turn is 3/8.

(b) The probability that Valeria has three heads on her first turn is (1/2) * (1/2) * (1/2) = 1/8, since each coin has a probability of 1/2 of coming up heads and all three coins are flipped.

(c) The probability that Valeria has two heads after her second turn is the probability that she had one tail and two heads on her first turn, and then flipped the tail on her second turn. There are three ways that she could have had one tail and two heads on her first turn: the first coin was tails and the other two were heads, the second coin was tails and the other two were heads, or the third coin was tails and the other two were heads. In each of these cases, the probability of flipping the tail on the second turn is 1/2. Therefore, the probability that Valeria has two heads after her second turn is 3/8 * 1/2 = 3/16.

(d) The probability that Valeria has one head after her second turn is the probability that she had two tails and one head on her first turn, and then flipped one of the tails on her second turn. There are three ways that she could have had two tails and one head on her first turn: the first coin was heads and the other two were tails, the second coin was heads and the other two were tails, or the third coin was heads and the other two were tails. In each of these cases, the probability of flipping one of the tails on the second turn is 1/2. Therefore, the probability that Valeria has one head after her second turn is 3/8 * 1/2 = 3/16.

(e) The probability that Valeria has one head after her third turn is the probability that she had two tails and one head on her first turn, flipped the head and one of the tails on her second turn, and then flipped the remaining tail on her third turn. There are three ways that she could have had two tails and one head on her first turn, and we can consider each case separately:

- If the first coin was heads and the other two were tails, then the probability of flipping the head and one of the tails on the second turn is 1/2, and the probability of flipping the remaining tail on the third turn is 1/2. Therefore, the probability of this sequence of flips is (1/2) * (1/2) * (1/2) = 1/8.
- If the second coin was heads and the other two were tails, then the probability of flipping the head and one of the tails on the second turn is 1/2, and the probability of flipping the remaining tail on the third turn is 1/2. Therefore, the probability of this sequence of flips is (1/2) * (1/2) * (1/2) = 1/8.
- If the third coin was heads and the other two were tails, then the probability of flipping the head and one of the tails on the second turn is 1/2, and the probability of flipping the remaining tail on the third turn is 1/2. Therefore, the probability of this sequence of flips is (1/2) * (1/2) * (1/2) = 1/8.

Adding up the probabilities for each case, we get a total probability of 3/8 * 1/8 = 3/64.

(f) The probability that Valeria has three heads after her fourth turn is the probability that she had one tail and two heads on her first turn, flipped the tail on her second turn, flipped the two tails on her third turn, and flipped the remaining head on her fourth turn. There is only one way that she could have had one tail and two heads on her first turn, and the probability of this is 3/8. The probability of flipping the tail on the second turn is 1/2. The probability of flipping the two tails on the third turn is 1/4, since there are two tails that need to be flipped and each has a probability of 1/2 of coming up tails. Finally, the probability of flipping the remaining head on the fourth turn is 1/2.

Mar 13, 2023