Jar A contains 8 red balls and 2 green balls. Jar B contains 6 red balls and 3 green balls. You randomly select one ball from Jar A and, without knowing its color, drop it into Jar B. You then randomly select one ball from Jar B.
What is the probability that the ball you select from Jar B will be red?
Before the change:
Jar A contains 8 red balls and 2 green balls.
Jar B contains 6 red balls and 3 green balls.
After the change:
Jar B contains 6 red balls, 3 green balls,
and one ball that has a 1/5 chance of being green and 4/5 chance of being red.
So, total balls in jar B = 10
While we consider a normal red ball to be 'one red ball' we can take the unknown ball to be '4/5 red ball'.
So, total red balls = 6 + 4/5 = 34/5
Probability of red ball = Red balls/Total balls
= 34/5 x 1/10 = 34/50
= 0.68
Another way to look at it:
Jar A has 8R and 2G, total=10
Jar B has 6R and 3G, total=9
We transfer 1 ball from A to B.
8 time out of 10, we transferred a red, so B contains 7R+3G, total=10
Probability of drawing a red
= 7/10 (8 times out of 10)
= 7/10 * (8/10)
=0.56
2 times out of 10, we transfer a green, so B contains 6R+4G, total=10
Probability of drawing a red
= 6/10 *(2/10)
=0.12
Total probability
=0.56+0.12
=0.68
To find the probability that the ball selected from Jar B will be red, we need to consider the different possible scenarios.
There are two scenarios to consider:
1. The ball selected from Jar A is red.
2. The ball selected from Jar A is green.
Let's calculate the probability for each scenario:
1. The ball selected from Jar A is red:
In Jar A, there are 8 red balls and 2 green balls. The probability of selecting a red ball from Jar A is therefore 8/10 = 4/5.
After placing the red ball from Jar A into Jar B, the total number of balls in Jar B becomes 10 (6 original red balls + 3 original green balls + 1 red ball from Jar A).
Therefore, the probability of selecting a red ball from Jar B, given the ball from Jar A is red, is 7/10.
2. The ball selected from Jar A is green:
In Jar A, there are 8 red balls and 2 green balls. Therefore, the probability of selecting a green ball from Jar A is 2/10 = 1/5.
After placing the green ball from Jar A into Jar B, the total number of balls in Jar B becomes 10 (6 original red balls + 3 original green balls + 1 green ball from Jar A).
Therefore, the probability of selecting a red ball from Jar B, given the ball from Jar A is green, is 6/10 = 3/5.
To find the overall probability of selecting a red ball from Jar B, we need to consider both scenarios:
Probability = (Probability of selecting a red ball from Jar A) * (Probability of selecting a red ball from Jar B, given the ball from Jar A is red)
+ (Probability of selecting a green ball from Jar A) * (Probability of selecting a red ball from Jar B, given the ball from Jar A is green)
= (4/5 * 7/10) + (1/5 * 3/5)
= 28/50 + 3/25
= (28 * 5 + 3) / (50 * 5)
= 140 + 3 / 250
= 143 / 250
Therefore, the probability that the ball selected from Jar B will be red is 143/250.
To determine the probability of selecting a red ball from Jar B, we need to consider the possible scenarios and their respective probabilities.
Let's break down the process step by step:
Step 1: Selecting a ball from Jar A
Jar A contains a total of 8 red balls and 2 green balls, making a total of 10 balls.
The probability of selecting a red ball from Jar A is given by:
Probability of selecting a red ball from Jar A = Number of red balls in Jar A / Total number of balls in Jar A
Probability of selecting a red ball from Jar A = 8 / 10 = 0.8
Similarly, the probability of selecting a green ball from Jar A is given by:
Probability of selecting a green ball from Jar A = Number of green balls in Jar A / Total number of balls in Jar A
Probability of selecting a green ball from Jar A = 2 / 10 = 0.2
Step 2: Transferring the selected ball from Jar A to Jar B
After transferring the selected ball from Jar A to Jar B, the number of balls in Jar B becomes:
Total number of balls in Jar B = Total number of balls in Jar A + Total number of balls in Jar B
Total number of balls in Jar B = 10 + 9 = 19
Step 3: Selecting a ball from Jar B
Jar B now contains a total of 6 red balls (originally in Jar B) and 2 additional balls (1 red and 1 green) transferred from Jar A.
The probability of selecting a red ball from Jar B can be calculated as follows:
Probability of selecting a red ball from Jar B = Number of red balls in Jar B / Total number of balls in Jar B
Probability of selecting a red ball from Jar B = 6 / 19
Therefore, the probability of selecting a red ball from Jar B is 6/19 or approximately 0.3158 (to four decimal places).