1
$\begingroup$

If you Google “Binomial probability word questions,” you find a bunch of questions like this one: If a coin has a 50% chance of flipping H on a single flip, what’s the probability that it will do so in exactly 3 out of 4 flips? That question is typical in that the probability on a single trial is given, and a probability related to a set of trials is wanted.

What if we reverse that, giving the probability on a set of trials and asking the probability on a single trial? Is that solvable?

THE QUESTION:

Every day, Al and Bob play 5 games of pool. Then they go out for lunch, and the loser of the majority of the day’s games has to pay for the lunch. If Al has to pay for lunch 80% of the time, what is Al’s probability of winning a single game of pool? Explain your answer VERBALLY and with math notation.

MY ANSWER, SUCH AS IT IS:

Let p represent the probability that Al will win on a single trial. The complete probability distribution for 5 trials looks like this:

    x   nCx p^x (1-p)^(n-x) 
    0   5C0 p^0 (1-p)^5  = 1 p^0(1-p)^5
    1   5C1 p^1 (1-p)^4  = 5 p^1(1-p)^4
    2   5C2 p^2 (1-p)^3  = 10 p^2(1-p)^3
    3   5C3 p^3 (1-p)^2  = 10 p^3(1-p)^2
    4   5C4 p^4 (1-p)^1  = 5 p^4(1-p)^1
    5   5C5 p^5 (1-p)^0  = 1 p^5(1-p)^0

The probability that Al will win 0 or 1 or 2 games out of 5 is given to be 80%. So

1 p^0(1-p)^5 + 5 p^1(1-p)^4 +10 p^2(1-p)^3 = .8

Then what?

  • 0
    Then solve this equation for $p\in(0,1)$. You'll get the probability of winning one game for Al. – 2017-01-11
  • 0
    You’ve got a fifth-degree polynomial equation in $p$. All you have to do now is solve it. Graphing it shows that there’s only one real root. – 2017-01-11
  • 0
    I understand amd's suggestion of graphing, but I wonder if there is no computational solution. Both of these answers intend graphing when they use the word "solve"? – 2017-01-12

0 Answers 0