- Thread starter
- #1

#### OhMyMarkov

##### Member

- Mar 5, 2012

- 83

Hello everyone!

I'm confusing conditional and joint probability again, here is the problem:

Given 4 bottles of milk, one of them is sour. We need to find out which of the bottles is sour by tasting the least number of bottles. By inspection, we can roughly tell the probability of every bottle being sour. Therefore we rank the bottles in non-ascending order of probability that the bottle is sour. Assume that these probabilities are $p_1, p_2, p_3,$ and $p_4$.

We also define an RV $X$ that is equal to the number of bottles we taste before determining what bottle is sour: $X\in\{1,2,3\}$.

Here is where I'm confused:

$P\{X=1\}=P\{b_1 \text{bad}\}=p_1$

Now, which of the following is correct?

(1) $P\{X=2\}=P\{b_2 \text{bad} \; | \; b_1 \text{good}\}$, $P\{X=3\}=P\{b_3 \text{bad} \;|\; b_1, b_2 \text{good}\}$

(2) $P\{X=2\}=P\{b_2 \text{bad and} \; b_1 \text{good}\}$, $P\{X=3\}=P\{b_3 \text{bad} \; \text{and} \; b_1 \text{good and} \; b_2 \text{good}\}$

Thanks!

I'm confusing conditional and joint probability again, here is the problem:

Given 4 bottles of milk, one of them is sour. We need to find out which of the bottles is sour by tasting the least number of bottles. By inspection, we can roughly tell the probability of every bottle being sour. Therefore we rank the bottles in non-ascending order of probability that the bottle is sour. Assume that these probabilities are $p_1, p_2, p_3,$ and $p_4$.

We also define an RV $X$ that is equal to the number of bottles we taste before determining what bottle is sour: $X\in\{1,2,3\}$.

Here is where I'm confused:

$P\{X=1\}=P\{b_1 \text{bad}\}=p_1$

Now, which of the following is correct?

(1) $P\{X=2\}=P\{b_2 \text{bad} \; | \; b_1 \text{good}\}$, $P\{X=3\}=P\{b_3 \text{bad} \;|\; b_1, b_2 \text{good}\}$

(2) $P\{X=2\}=P\{b_2 \text{bad and} \; b_1 \text{good}\}$, $P\{X=3\}=P\{b_3 \text{bad} \; \text{and} \; b_1 \text{good and} \; b_2 \text{good}\}$

Thanks!

Last edited: