A box contains 5 yellow balls, 3 blue, and 1 red ball. Two balls are drawn at random. Find the probability that:
The two balls have the same color
you could pick either two yellow or 2 blue
prob(your event) = (5/9)(4/8) + (3/9)(2/8)
= 5/18 + 1/12
= 13/36
or ( C(5,2) + C(3,2) )/C(9,2)
= same as above
To find the probability that the two balls have the same color, we need to determine the total number of possible outcomes and the number of favorable outcomes.
Total number of outcomes:
When two balls are drawn, there are a total of 9 balls in the box. So, the total number of outcomes is given by choosing any 2 balls out of the 9, which is calculated as C(9,2) or 9C2, where C stands for combination. The formula for C(n,r) is n! / (r!(n-r)!), where n is the total number of items and r is the number of items chosen.
C(9,2) = 9! / (2!(9-2)!) = 9! / (2!7!) = (9 * 8 * 7!) / (2! * 7!) = (9 * 8) / 2 = 36
Number of favorable outcomes:
For the two balls to have the same color, we can select both yellow balls (C(5,2)), both blue balls (C(3,2)), or both red balls (C(1,2)). Adding these three cases together will give us the number of favorable outcomes.
C(5,2) = 5! / (2!(5-2)!) = 5! / (2!3!) = (5 * 4 * 3!) / (2! * 3!) = (5 * 4) / 2 = 10
C(3,2) = 3! / (2!(3-2)!) = 3! / (2!1!) = (3 * 2 * 1!) / (2! * 1!) = 3
C(1,2) = 0 (Since there is only 1 red ball and we need to select 2)
Number of favorable outcomes = C(5,2) + C(3,2) + C(1,2) = 10 + 3 + 0 = 13
Now, we can calculate the probability by dividing the number of favorable outcomes by the total number of outcomes:
Probability = Number of favorable outcomes / Total number of outcomes = 13 / 36 = 0.3611 (rounded to four decimal places)
Therefore, the probability that the two balls drawn have the same color is approximately 0.3611 or 36.11%.
To find the probability that the two balls have the same color, we need to calculate the number of favorable outcomes (same color) and the number of possible outcomes.
Number of favorable outcomes:
There are three colors in the box: yellow, blue, and red.
To calculate the number of favorable outcomes, we need to consider the possible combinations of colors.
1) Yellow: We can choose 2 balls out of the 5 yellow balls.
Number of combinations = C(5, 2) = 5! / (2! * (5-2)!) = 10
2) Blue: We can choose 2 balls out of the 3 blue balls.
Number of combinations = C(3, 2) = 3! / (2! * (3-2)!) = 3
3) Red: We can choose 2 balls out of the 1 red ball.
Number of combinations = C(1, 2) = 1! / (2! * (1-2)!) = 0 (as there is only 1 red ball, we cannot choose 2 balls of the same color)
Total number of favorable outcomes = 10 + 3 + 0 = 13
Number of possible outcomes:
We need to choose 2 balls out of the total 9 balls in the box.
Number of combinations = C(9, 2) = 9! / (2! * (9-2)!) = 36
Probability of drawing two balls of the same color = Number of favorable outcomes / Number of possible outcomes
= 13 / 36
Therefore, the probability that the two balls have the same color is 13/36.