# What is the probability of picking 5 red balls when 5 balls are picked randomly in the following scenario?A box has 20 balls, 5 of which are red, 5 are blue, 5 are green and 5 are white. A ball is...

What is the probability of picking 5 red balls when 5 balls are picked randomly in the following scenario?

A box has 20 balls, 5 of which are red, 5 are blue, 5 are green and 5 are white. A ball is picked and kept aside if it is red but returned if it is not red. What is the probability of picking 5 red balls when 5 balls are picked randomly?

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### 2 Answers

We initially have 20 balls of which 5 are red in color.

Now when the first ball is picked, the probability of it being red is 5/20 = 1/4.

If the first ball picked was red, we now have 19 balls and 4 red balls. Therefore the probability of picking a red ball is 4/19 and the probability of the first two balls being red is (1/4)*(4/19)

Now if the first two balls picked are red we have 18 balls of which 3 are red in color. Therefore the probability of picking a red ball is 3/18. The probability of picking 3 red balls in a row is (1/4)*(4/19)*(3/18)

If the first three balls were red we are left with 17 balls of which 2 are red in color. Therefore the chance of picking a red ball is 2/17. This makes the probability of picking 4 red balls in a row equal to (1/4)*(4/19)*(3/18)*(2/17).

If all the 4 balls picked initially are red, we are left with 16 balls one of which is red. The probability of picking a red ball now is 1/16.

This makes the probability of picking all 5 red balls as (1/4)*(4/19)*(3/18)*(2/17)*(1/16) = 1/15504

**Therefore the probability of picking 5 red balls when 5 balls are picked randomly is 1/ 15504.**

There are 20 balls. Of these 5 are red.

In the first draw , the probablity of that the ball drawn is red is 5/20

Having the drawn the first ball , there are 19 balls and 4 red balls. Therefore the probablity of drawing a red ball in is 4/19.

Similarly the probability of drawing a 3rd red ball after having drawn 2 red balls is (5-2)/20-2) = 3/18.

Like that probability if drawing the red ball out of 17 remaining balls is 2/17.

The probability of drawing the red ball after having drawn 4 red balls earlier in the 5th draw is 1/16.

Therefore in the first 5 draws the probability that 5 red balls are drawn = 5/20*4/19*3/18*2/117*1/16 = 5!/(20*19*18*17*16) = 120/(20*19*18*17*16) = 1/(19*3*17*16) = 1/15504 = 0.0000645 nearly.