you are given 9 to 1 odds aganist tossing three heads with three coins, meaning you win $9 if you succeed and you lose a $1 if you fail
What is your question?
To determine the probability of tossing three heads with three coins, we need to calculate the probability of a single coin landing on heads and then raised to the power of three since there are three coins.
The probability of a single coin landing on heads is 1/2, assuming a fair coin. Therefore, the probability of getting heads three times in a row is (1/2)^3 = 1/8.
Now, let's calculate the expected value. The expected value is obtained by multiplying each outcome by its respective probability and summing them.
For winning $9, the probability is 1/8 and the value is $9. So, the contribution to the expected value is (1/8) * $9 = $1.125.
For losing $1, the probability is 7/8 (since it's the complement of winning) and the value is -$1. So, the contribution to the expected value is (7/8) * -$1 = -$0.875.
The expected value is the sum of these contributions: $1.125 + (-$0.875) = $0.25.
Since the expected value is positive ($0.25), it means that on average, you would expect to win $0.25 for each game played. Therefore, it would be beneficial to play this game in the long run.
However, it's important to note that the odds of winning or losing in a single game are independent of previous outcomes. So, each time you play, the probability of winning remains 1/8, and you may experience different results in each game.