Suppose a box contains 1500 balls out of which 7500 are white and the remaining are black balls. if we draw a sample 8 balls randomly (with out replacement). What is the probability that they are all white ball?
We can use the hypergeometric distribution to solve this problem.
The total number of balls in the box is N = 1500, out of which n = 7500 are white and N - n = 7500 are black. We want to know the probability of drawing a sample of k = 8 balls that are all white without replacement.
The probability of drawing the first white ball is n/N = 7500/1500 = 0.5.
The probability of drawing the second white ball, without replacement, is (n-1)/(N-1) = 7499/1499.
Following the same reasoning, the probability of drawing 8 white balls, without replacement, is:
P(8 white balls) = (n/N) * ((n-1)/(N-1)) * ((n-2)/(N-2)) * ... * ((n-7)/(N-7))
= (7500/1500) * (7499/1499) * (7498/1498) * ... * (7493/1493)
= 0.0444 (rounded to four decimal places)
Therefore, the probability of drawing a sample of 8 balls that are all white without replacement from a box containing 1500 balls of which 7500 are white is 0.0444.
To find the probability that all 8 balls drawn are white, we need to calculate the number of ways we can choose 8 white balls out of the total number of ways we can choose any 8 balls from the box.
First, let's calculate the total number of ways to choose 8 balls from the box without replacement:
Total number of ways to choose 8 balls out of 1500 = C(1500, 8)
where C(n, r) represents the combination of n items taken r at a time.
Next, let's calculate the number of ways to choose 8 white balls out of the 750 white balls in the box:
Number of ways to choose 8 white balls out of 750 = C(750, 8)
Finally, we can calculate the probability:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability of drawing all white balls = C(750, 8) / C(1500, 8)
Using the combination formula C(n, r) = n! / (r! * (n-r)!), we can calculate this probability.