Answer to Question #323696 in Statistics and Probability for Klum

Question

From a pocket containing 10 coins (20 cents each) and 10 coins (3 cents each), 10 coins are taken randomly. What is the probability that the amount of money left in the pocket is not less than the one taken out

Answer 1

Total cost of coins is 10x20+10x3=230

Half of this value is 230/2=115. So we should take less than 115 coins.

115/20=5.75. So we can take maximum 5 coins ( 20 cents).

Coins (3 cents we can take from 5(10-5) to 10)

0 (20 cents)+10 (3 cents)

At the start there are 10 (3 cent) coins. When we take 10 times (3 cent) coins from pocket, probabilities are 10/20,(10-1=9)/(20-1)=9/19, and so on. Last is 1/11. Then we should multiply these probabilities.

$P=\frac{10!}{20x19x18x17x16x15x14x13x12x11}=0.000005$

1 (20 cents)+9 (3 cents)

Probability of 1 (20 cents) is 10/20, then of 9 times of (3 cents) are 10/19, 9/18, and so on. Last is 2/11.

$P=\frac{10x9x8x7x6x5x4x3x2x10}{20x19x18x17x16x15x14x13x12x11}=0.000054$

2 (20 cents)+8 (3 cents)

Probabilities of 2 (20 cents) are 10/20 and 9/19 then of 8 times of (3 cents) are 10/18, 9/17, and so on. Last is 3/11.

$P=\frac{10x9x8x7x6x5x4x3x10x9}{20x19x18x17x16x15x14x13x12x11}=0.000244$

3 (20 cents)+7 (3 cents)

Probabilities of 3 (20 cents) are 10/20 and 9/19, 8/18 then of 7 times of (3 cents) are 10/17, 9/16, and so on. Last is 4/11.

$P=\frac{10x9x8x7x6x5x4x10x9x8}{20x19x18x17x16x15x14x13x12x11}=0.000650$

4 (20 cents)+6 (3 cents)

Probabilities of 4 (20 cents) are 10/20 and 9/19, 8/18, 7/17 then of 6 times of (3 cents) are 10/16, 9/15, and so on. Last is 5/11.

At the start$P=\frac{10x9x8x7x6x5x10x9x8x7}{20x19x18x17x16x15x14x13x12x11}=0.001137$

5 (20 cents)+5 (3 cents)

Probabilities of 5 (20 cents) are 10/20 and 9/19 and so on. Last is 6/16, then of 5 times of (3 cents) are 10/15, 9/14, and so on. Last is 6/11.

• $P=\frac{10x9x8x7x6x10x9x8x7x6}{20x19x18x17x16x15x14x13x12x11}=0.001364$

Probability that amount of money left in the pocket is not less than the one taken out is 0.000005+0.000054+0.000244+0.000650+0.001137+0.001364=0.003454