A bag contains different colored balls; 7 yellow, 3 red, and 6 green. I randomly select one ball, note its color, leave it out of the bag, and select another ball. What is the probability that the first ball is red and the second is yellow, P(RY)?
To find the probability that the first ball is red and the second ball is yellow, P(RY), we need to calculate the probability of two events happening consecutively.
Step 1: Determine the probability of selecting a red ball first.
There are a total of 7 yellow balls, 3 red balls, and 6 green balls in the bag. The probability of selecting a red ball first is:
P(R) = Number of red balls / Total number of balls
P(R) = 3 / (7 + 3 + 6)
P(R) = 3 / 16
Step 2: Once a red ball is selected and removed from the bag, the total number of balls is reduced to 15. We need to determine the probability of selecting a yellow ball out of the remaining 15 balls.
There are now 6 yellow balls remaining in the bag. The probability of selecting a yellow ball is:
P(Y) = Number of yellow balls / Total number of remaining balls
P(Y) = 6 / 15
Step 3: Calculate the probability of both events occurring consecutively by multiplying the individual probabilities:
P(RY) = P(R) * P(Y)
P(RY) = (3/16) * (6/15)
P(RY) = 18/240
P(RY) = 3/40
Therefore, the probability that the first ball is red and the second ball is yellow, P(RY), is 3/40.
To calculate the probability of selecting a red ball followed by a yellow ball, we need to determine the total number of possible outcomes and the number of favorable outcomes.
Total number of balls in the bag: 7 yellow + 3 red + 6 green = 16 balls
When we take out one ball and select another ball from the remaining 15, the total number of possible outcomes in this scenario is 16 * 15 = 240.
Now, let's determine the number of favorable outcomes, i.e., when the first ball is red and the second ball is yellow.
Number of favorable outcomes:
Since we have already taken out one ball and noted its color, we are left with 15 balls in the bag.
The number of favorable outcomes is selecting 1 red ball out of 3 red balls and 1 yellow ball out of 7 yellow balls, which can be calculated as 3 * 7 = 21.
Therefore, the probability of selecting a red ball followed by a yellow ball, P(RY), is given by:
P(RY) = Number of favorable outcomes / Total number of possible outcomes
P(RY) = 21 / 240
P(RY) ≈ 0.0875 or 8.75%