a bag contains 14 identical balls,3of which are red, 4black and 7 white .5 balls are drawn from the bag .find the probability that[1] are red [2] at least 3 are black
To find the probability of drawing certain colored balls from a bag, we need to determine the total number of possible outcomes and the number of favorable outcomes.
Let's calculate the probability for each part of the question:
1. Probability of drawing 1 red ball:
First, we need to calculate the total number of ways to draw 5 balls out of 14. This can be done using combination formula C(n, r), which gives the number of ways to choose r items from a set of n distinct items. So, in this case, n = 14 (total balls) and r = 5 (balls drawn).
Total possible outcomes = C(14, 5) = 14! / (5! * (14-5)!) = 2002
Now, we need to determine the number of favorable outcomes, which is drawing exactly 1 red ball. There are 3 red balls in the bag, and we need to choose 1 out of those 3.
Number of favorable outcomes = C(3, 1) = 3
Now, we can calculate the probability of drawing 1 red ball by dividing the number of favorable outcomes by the total possible outcomes:
Probability of drawing 1 red ball = Number of favorable outcomes / Total possible outcomes
= 3 / 2002
≈ 0.0015 (rounded to four decimal places)
So, the probability of drawing exactly 1 red ball is approximately 0.0015.
2. Probability of drawing at least 3 black balls:
To find the probability of drawing at least 3 black balls, we need to calculate the number of favorable outcomes for drawing 3 black balls, 4 black balls, and 5 black balls individually, and then sum them up.
Number of favorable outcomes for drawing exactly 3 black balls = C(4, 3) = 4
Number of favorable outcomes for drawing exactly 4 black balls = C(4, 4) = 1
Number of favorable outcomes for drawing exactly 5 black balls = C(4, 5) = 0 (since there are only 4 black balls available)
So, the total number of favorable outcomes for drawing at least 3 black balls = 4 + 1 + 0 = 5
Now, we need to calculate the total possible outcomes, which is the same as the previous calculation: C(14, 5) = 2002.
Finally, we can calculate the probability of drawing at least 3 black balls by dividing the number of favorable outcomes by the total possible outcomes:
Probability of drawing at least 3 black balls = Number of favorable outcomes / Total possible outcomes
= 5 / 2002
≈ 0.0025 (rounded to four decimal places)
Therefore, the probability of drawing at least 3 black balls is approximately 0.0025.