There are 15 balls in all, so
P(red,red) = 3/15 * 2/14 = 1/35
what do you get for the others?
P(red,red) = 3/15 * 2/14 = 1/35
what do you get for the others?
Let's start with Part (a), finding the probability that both balls are red:
Step 1: Calculate the number of favorable outcomes.
Since there are 3 red balls in the basket, the number of ways to choose 2 red balls is given by the combination formula: C(3, 2) = 3.
Step 2: Calculate the total number of possible outcomes.
We need to choose 2 balls from a total of 3 red balls, 5 blue balls, and 7 green balls. So, the total number of possible outcomes is given by the combination formula: C(15, 2) = 105.
Step 3: Calculate the probability.
Probability = Number of favorable outcomes / Total number of possible outcomes
P(Both are red) = 3/105 = 1/35
So, the probability that both balls are red is 1/35.
Moving on to Part (b), finding the probability that the first is blue and the other is green:
Step 1: Calculate the number of favorable outcomes.
We have 5 blue balls to choose from and 7 green balls to choose from for the second pick. So, the number of favorable outcomes is 5 * 7 = 35.
Step 2: Calculate the total number of possible outcomes.
We need to choose 2 balls from a total of 3 red balls, 5 blue balls, and 7 green balls. So, the total number of possible outcomes is C(15, 2) = 105.
Step 3: Calculate the probability.
P(First is blue, the other is green) = 35/105 = 1/3
So, the probability that the first ball is blue and the other is green is 1/3.
Let's move on to Part (c), finding the probability that one is blue and the other is green:
Step 1: Calculate the number of favorable outcomes.
Similar to Part (b), the number of ways to choose one blue ball from 5 and one green ball from 7 is 5 * 7 = 35.
Step 2: Calculate the total number of possible outcomes (same as Parts a and b).
The total number of possible outcomes is C(15, 2) = 105.
Step 3: Calculate the probability.
P(One is blue, the other is green) = 35/105 = 1/3
So, the probability that one ball is blue and the other is green is 1/3.
Moving on to Part (d), finding the probability that both balls are of different colors:
Step 1: Calculate the number of favorable outcomes.
We need to choose one ball from 5 blue balls and one ball from 7 green balls. So, the number of favorable outcomes is 5 * 7 = 35.
Step 2: Calculate the total number of possible outcomes.
The total number of possible outcomes is C(15, 2) = 105 (same as Parts a, b, and c).
Step 3: Calculate the probability.
P(Both are of different colors) = 35/105 = 1/3
So, the probability that both balls are of different colors is 1/3.
Finally, let's move on to Part (e), finding the probability that both balls are of the same color:
Step 1: Calculate the number of favorable outcomes for each color.
For red balls, there are C(3, 2) = 3 favorable outcomes.
For blue balls, there are C(5, 2) = 10 favorable outcomes.
For green balls, there are C(7, 2) = 21 favorable outcomes.
Step 2: Calculate the total number of possible outcomes.
The total number of possible outcomes is C(15, 2) = 105 (same as Parts a, b, c, and d).
Step 3: Calculate the probability.
P(Both are of the same color) = (3 + 10 + 21)/105 = 34/105
So, the probability that both balls are of the same color is 34/105.