+82 Valeria has three coins, each of which has a probability of 2/3 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 on her first turn.
(a) What is the probability that Valeria obtains all heads on her first turn?
(b) What is the probability that Valeria obtains all heads on her first turn or her second turn?
+1199 (a)
The probability that Valeria obtains all heads on her first turn is the product of the probabilities that each individual coin comes up heads, which is (32)3=278.
(b)
To find the probability that Valeria obtains all heads on her first turn or her second turn, we need to consider two cases:
Case 1: Valeria obtains all heads on her first turn.
Case 2: Valeria does not obtain all heads on her first turn, but she does obtain all heads on her second turn.
The probability of Case 1 is278, as we found in part (a).
To calculate the probability of Case 2, we need to consider the following steps:
Valeria flips all three coins on her first turn.
At least one of the coins comes up tails.
Valeria flips any coins that came up tails on her first turn on her second turn.
All of the coins come up heads on her second turn.
The probability of each of these steps is as follows:
Step 1: Always happens, so the probability is 1.
Step 2: The probability of at least one of the coins coming up tails is 1−(32)3=2719.
Step 3: Valeria flips the coins that came up tails on her first turn, so the probability of this step is 1.
Step 4: The probability of all of the coins coming up heads on her second turn is (32)3=278.
Therefore, the probability of Case 2 is:
\frac{19}{27} \cdot 1 \cdot 1 \cdot \frac{8}{27} = \frac{152}{729}
The probability that Valeria obtains all heads on her first turn or her second turn is the sum of the probabilities of Case 1 and Case 2, which is:
\frac{8}{27} + \frac{152}{729} = \boxed{\frac{192}{729}}
