Probability question

[Vintage]

lets say you have a 1% chance of winning a game. You play the game 100 times. What are the odds that you will win the game at least once?

similar question: you have a 5% chance of winning a game.... you play the game 20 times. What are the odds you win at least once?

[uuDevil]

[ QUOTE ]
lets say you have a 1% chance of winning a game. You play the game 100 times. What are the odds that you will win the game at least once?

[/ QUOTE ]

Assuming independent trials, this is

1-P(losing 100 times in a row)=
1-(.99)^100= .634
or odds of 0.577:1.

You can do the 2nd question the same way.

[pzhon]

(1- 1/n)^n is about 1/e, where e is Euler's number 2.7182818... More precisely, it is about 1/e(1-1/(2n)). You can get this from the binomial theorem and the power series e^x = 1+x+x^2/2+x^3/6+x^4/24...

For both situations, the probability of losing all games is about 1/e, so the chance of winning at least once is about 1-1/e ~ 63.21%. The exact values are 63.40% for 100 trials with 1% success rate and 64.15% for 20 trials with 5% success rate.