? 發問於 科學及數學數學 · 1 十年前

# Expected gain

A carnival sharpshooter game charges \$25 for 25 shots at a target.

If the shooter hits the bullseye fewer than 5 times then he gets no prize.

If he hits the bullseye 5 times he gets back \$10.

For each additional bullseye over 5 he gets back an additional \$5.

The shooter estimates that he has a 0.2 probability of hitting the bullseye on any given shot.

What is the shooter&#39;s expected gain if he plays the game ?

### 1 個解答

• Karin
Lv 6
1 十年前
最愛解答

the shooter's expected gain

= 25C5x(0.2)^5x(0.8)^20x(10)

+ 25C6x(0.2)^6x(0.8)^19x(10+5)

+ 25C7x(0.2)^7x(0.8)^18x(10+2x5)

+25C8x(0.2)^8x(0.8)^17x(10+3x5)

+ ...

+ 25C25x(0.2)25x(0.8)^0x(10+20x5)

= \$9.7136

2006-11-22 09:04:01 補充：

\$9.7136 is the expected gain after giving the charges.If the charges is included, the expected gain will be = -\$ 25 + \$9.7136= \$15.2864 = \$15.29 (2 sig. figures)

2006-11-22 09:04:42 補充：

More Explanations:For hitting 5 bullseye out of 25, combination is 25C5, chance is 25C5 x (0.2)^5 x (0.8)^20,gain is 25C5 x (0.2)^5 x (0.8)^20 x (10)

2006-11-22 09:05:13 補充：

For hitting 6 bullseye out of 25, combination is 25C6, chance is 25C6 x (0.2)^6 x (0.8)^19,gain is 25C6 x (0.2)^6 x (0.8)^19 x (15)and so on,

2006-11-22 09:05:25 補充：

For hitting 25 bullseye out of 25, combination is 25C25, chance is 25C25 x (0.2)^25 x (0.8)^0,gain is 25C25 x (0.2)^25 x (0.8)^0 x (110)The total return is the sum of all these gains.The expected gain is the total return - amount paid.

2006-11-22 09:06:34 補充：

Corrections:The answer should be= -\$ 25 + \$9.7136= -\$15.2864 = -\$15.29 (2 sig. figures)