Question

Contestants on a gameshow spin a wheel with \[24\] equally-sized segments. most of those segments show different prize amounts, but \[2\] of them are labeled "bankrupt": suppose that a contestant is going to spin the wheel twice in a row. here are some events and their meanings: event meaning \[b_1\] the first spin lands on bankrupt. \[b_2\] the second spin lands on bankrupt. \[b^c_1\] the first spin does not land on bankrupt. \[b^c_2\] the second spin does not land on bankrupt. consider this probability: \[p \left( b_1 \text{ and } b_2 \right) = p(b_1) \cdot p(b_2 \mid b_1)\] what does \[p \left( b_2 \mid b_1 \right)\] represent in this context?

Answer 1

\( p(b_2 \mid b_1) \) represents the probability that the second spin lands on "bankrupt", which is the same as the probability of landing on "bankrupt" for any spin, \(\frac{1}{12}\), as the spins are independent of each other.

Step-by-step explanation:

In the context of this probability question, \( p(b_2 \mid b_1) \) represents the probability that the second spin lands on "bankrupt" given that the first spin has already landed on "bankrupt". Since each spin of the wheel is independent of the previous spins, the result of the first spin does not influence the result of the second spin.

To clarify this concept, let's go through this step by step.

1. The total number of segments on the wheel is 24.
2. Out of these segments, 2 are labeled "bankrupt".

The probability that any given spin lands on "bankrupt" is calculated as the number of "bankrupt" segments divided by the total number of segments. This calculation is based on the assumption that the wheel has a uniform distribution of landing on any segment:

\[ p(bankrupt) = \frac{\text{Number of "bankrupt" segments}}{\text{Total number of segments}} = \frac{2}{24} = \frac{1}{12} \]

Now, when we compute the probability \( p(b_2 \mid b_1) \), we are looking for the chance of spinning "bankrupt" on the second spin given that the first spin is already "bankrupt". Because the spins are independent, the first spin being "bankrupt" does not affect the probability of the second spin being "bankrupt". Therefore:

\[ p(b_2 \mid b_1) = p(bankrupt) \]

And we already know that \( p(bankrupt) = \frac{1}{12} \), so:

\[ p(b_2 \mid b_1) = \frac{1}{12} \]

In conclusion, \( p(b_2 \mid b_1) \) represents the probability that the second spin lands on "bankrupt", which is the same as the probability of landing on "bankrupt" for any spin, \(\frac{1}{12}\), as the spins are independent of each other.

Answer 2

In this context,
`\[p \left( b_2 \mid b_1 \right)\]` represents the probability of the second spin landing on bankrupt given that the first spin landed on bankrupt, and it is equal to 2/24.

Step-by-step explanation:

In this context,
`\[p \left( b_2 \mid b_1 \right)\]` represents the probability of the second spin landing on bankrupt given that the first spin landed on bankrupt.

When we calculate conditional probabilities, we consider the information or condition provided by the event that occurred earlier. In this case, the condition is that the first spin landed on bankrupt
`(\[b_1\])`. The probability of the second spin landing on bankrupt
`(\[b_2\])` given that the first spin already landed on bankrupt is denoted as
`\[p \left( b_2 \mid b_1 \right)\]`.

To calculate this probability, we need to determine the number of bankrupt segments on the wheel and the total number of segments. We are given that there are 24 equally-sized segments on the wheel, and 2 of them are labeled "bankrupt".

Since the wheel is spun twice, we assume that each spin is independent of the other. Therefore, the probability of landing on bankrupt for each spin remains the same.

The probability of the first spin landing on bankrupt
`(\[p(b_1)\])` is 2/24, as there are 2 bankrupt segments out of 24 in total.

Now, the probability of the second spin landing on bankrupt
`(\[p(b_2 \mid b_1)\])` is also 2/24. This is because the outcome of the first spin does not affect the outcome of the second spin since each spin is independent.

Therefore, in this context,
`\[p \left( b_2 \mid b_1 \right)\]` represents the probability of the second spin landing on bankrupt given that the first spin landed on bankrupt, and it is equal to 2/24.