Three boxes, labeled A,.B. C, contain 1 red and 2 black balls, 2 red and 1 black balls, and 1 red and 1 black ball, respectively. Compute the terminal probabilities by using the multiplication technique. Then add the terminal probabilities for all the outcomes in which the ball is red.
To compute the terminal probabilities using the multiplication technique, we need to find the probability of each event in a sequence happening. Let's label the events as follows:
Event 1: Picking box A
Event 2: Picking a red ball from box A
Event 3: Picking box B
Event 4: Picking a red ball from box B
Event 5: Picking box C
Event 6: Picking a red ball from box C
Now, we'll calculate the probabilities of each event happening:
Event 1: Picking box A
There are three boxes in total, so the probability of picking box A is 1/3.
Event 2: Picking a red ball from box A
Box A contains 1 red and 2 black balls, so the probability of picking a red ball from box A is 1/3.
Event 3: Picking box B
After picking box A, there are two boxes remaining, so the probability of picking box B is 1/2.
Event 4: Picking a red ball from box B
Box B contains 2 red and 1 black balls, so the probability of picking a red ball from box B is 2/3.
Event 5: Picking box C
After picking box B, there is only one box remaining, so the probability of picking box C is 1/1 or simply 1.
Event 6: Picking a red ball from box C
Box C contains 1 red and 1 black ball, so the probability of picking a red ball from box C is 1/2.
To calculate the terminal probability, we multiply the probabilities of the individual events:
Terminal probability = (1/3) * (1/3) * (1/2) * (2/3) * (1/1) * (1/2)
= 1/18
Now let's calculate the probability of picking a red ball. There are two possible outcomes where the ball is red: picking a red ball from box A and picking a red ball from box C.
For each outcome, we multiply the probabilities of the events leading to that outcome:
Outcome 1: Picking a red ball from box A
Probability = (1/3) * (1/3)
= 1/9
Outcome 2: Picking a red ball from box C
Probability = (1/3) * (1/2) * (1/1) * (1/2)
= 1/12
To find the total probability of picking a red ball, we add the probabilities of the individual outcomes:
Total probability = (1/9) + (1/12)
= 4/36 + 3/36
= 7/36
Therefore, the terminal probabilities for all the outcomes in which the ball is red is 7/36.