An urn holds 13 identical balls except that 2 are white, 4 are black, and 7 are red. An experiment consists of selecting two balls in succession without replacement and observing the color of each of the balls.
What is your question?
To solve this problem, we can use combinations and probability concepts. Let's break it down step by step:
Step 1: Calculate the total number of ways to choose two balls from the urn without replacement.
This can be done using the formula for combinations, which is given by nCr = n! / (r!(n-r)!), where n is the total number of balls and r is the number of balls being chosen. In this case, we want to choose 2 balls from a total of 13, so the number of ways to do this is 13C2 = 13! / (2!(13-2)!) = 13! / (2!11!) = (13 * 12) / (2 * 1) = 78.
Step 2: Calculate the probability of selecting two balls of a specific color.
We can calculate the probability of selecting two white balls, two black balls, and two red balls by counting the number of ways to choose balls of each color and dividing by the total number of ways to choose 2 balls.
- White balls: There are 2 white balls in the urn, so the number of ways to choose 2 white balls is 2C2 = 1. Thus, the probability of selecting 2 white balls is 1/78.
- Black balls: There are 4 black balls in the urn, so the number of ways to choose 2 black balls is 4C2 = 6. Thus, the probability of selecting 2 black balls is 6/78 = 1/13.
- Red balls: There are 7 red balls in the urn, so the number of ways to choose 2 red balls is 7C2 = 21. Thus, the probability of selecting 2 red balls is 21/78 = 7/26.
Step 3: Calculate the probability of selecting balls of different colors.
We can calculate the probability of selecting two balls of different colors by counting the number of ways to choose balls of different colors and dividing by the total number of ways to choose 2 balls.
- White and black balls: The number of ways to choose 1 white ball and 1 black ball is 2C1 * 4C1 = 2*4 = 8. Thus, the probability of selecting 1 white ball and 1 black ball is 8/78 = 4/39.
- White and red balls: The number of ways to choose 1 white ball and 1 red ball is 2C1 * 7C1 = 2*7 = 14. Thus, the probability of selecting 1 white ball and 1 red ball is 14/78 = 7/39.
- Black and red balls: The number of ways to choose 1 black ball and 1 red ball is 4C1 * 7C1 = 4*7 = 28. Thus, the probability of selecting 1 black ball and 1 red ball is 28/78 = 14/39.
And that's how you solve the problem by calculating probabilities using combinations.