each of three identical jewelry boxes has two drawers. in each drawer of the first box there is a gold watch. in each drawer of the second box there is a silver watch. in one drawer of the third box there is a gold watch while in the other there is a silver watch. if we select a box at random, open one of the drawers and find it to contain a silver watch, what is the probability that the other drawer has the gold watch?
detailed solution please
To solve this problem, we can use conditional probability. Let's break down the problem step by step to determine the probability of finding a gold watch in the other drawer given that we found a silver watch in one drawer.
Step 1: Understanding the scenario
We have three identical jewelry boxes, each having two drawers.
- In the first box, both drawers contain gold watches.
- In the second box, both drawers contain silver watches.
- In the third box, one drawer contains a gold watch, and the other contains a silver watch.
Step 2: Define the events
Let's define the events:
- A: Selecting the first box
- B: Selecting the second box
- C: Selecting the third box
- D: Opening a drawer and finding a silver watch
Step 3: Calculate the probabilities
Now, we need to determine the probability of selecting each box and finding a silver watch in one drawer.
The probability of selecting each box is 1/3 since there are three boxes available, and we select one randomly.
P(A) = P(B) = P(C) = 1/3
In the first box, both drawers have gold watches, so the probability of finding a silver watch (D) is zero.
P(D | A) = 0
In the second box, both drawers have silver watches, so the probability of finding a silver watch (D) is 1.
P(D | B) = 1
In the third box, one drawer has a gold watch, and one has a silver watch. If we randomly choose a drawer, there is a 50% chance (1/2) of selecting the silver watch drawer.
P(D | C) = 1/2
Step 4: Use Bayes' theorem to find the probability
Finally, we can use Bayes' theorem to calculate the probability of the other drawer having a gold watch (the event of interest) given that we found a silver watch in one drawer.
P(A | D) = (P(D | A) * P(A)) / (P(D | A) * P(A) + P(D | B) * P(B) + P(D | C) * P(C))
P(A | D) = (0 * 1/3) / (0 * 1/3 + 1 * 1/3 + 1/2 * 1/3)
P(A | D) = 0 / (0 + 1/3 + 1/6)
P(A | D) = 0 / (1/3 + 1/6)
P(A | D) = 0 / (2/6 + 1/6)
P(A | D) = 0 / 3/6
P(A | D) = 0
Therefore, the probability of the other drawer having a gold watch given that we found a silver watch in one drawer is 0.
To solve this problem, we can use conditional probability. Let's break it down step-by-step:
Step 1: Determine the probabilities of picking each box.
Since there are three identical jewelry boxes and we select one at random, the probability of picking any specific box is 1/3.
Step 2: Determine the probabilities of picking a silver watch from each box.
- Box 1: Since each drawer in Box 1 contains a gold watch, there is a 0% chance of picking a silver watch.
- Box 2: Since each drawer in Box 2 contains a silver watch, there is a 100% chance of picking a silver watch.
- Box 3: There is a 50% chance of picking a silver watch since one drawer contains a silver watch and the other contains a gold watch.
Step 3: Determine the total probability of picking a silver watch.
The total probability of picking a silver watch can be calculated by multiplying the probability of picking each box by the probability of picking a silver watch from that box, and then summing them up.
P(silver watch) = (1/3) * 0 + (1/3) * 1 + (1/3) * 0.5 = 1/3 + 1/3 + 1/6 = 1/2
Step 4: Determine the probability of the other drawer containing a gold watch given that we picked a silver watch.
Using conditional probability, we can now calculate the probability of the other drawer containing a gold watch (let's call this event A) given that we picked a silver watch (let's call this event B).
P(A|B) = (P(B|A) * P(A)) / P(B)
- P(B|A) represents the probability of picking a silver watch given that the other drawer contains a gold watch. It is equal to 1, since if the other drawer contains a gold watch, we are guaranteed to pick a silver watch.
- P(A) represents the probability of the other drawer containing a gold watch, which is 1/2 since one drawer contains a gold watch and the other contains a silver watch.
- P(B) represents the probability of picking a silver watch, which we calculated to be 1/2 in Step 3.
Plugging in the values, we get:
P(A|B) = (1 * 1/2) / (1/2) = 1
Therefore, the probability that the other drawer contains a gold watch given that we picked a silver watch is 1.