A fair coin is tossed 7 times. Let a/b be the probability of the event that at least 3 of the tosses are heads where an and b are positive and mutually prime. Find a+b

Answer 1

Let X be a random variable representing the number of heads out of 7 coin tosses.

Take the compliment of the given event, as it is faster and easier to calculate:

P(X >= 3) = 1 - P(X = 0, 1, or 2)

A coin toss is a Bernoulli trial, so 7 coin tosses would have a binomial distribution. Therefore, you can use the binomial formula to find P(X = 0, 1, or 2).

1 - ₇C₀(½)⁰(½)⁷ - ₇C₁(½)¹(½)⁶ - ₇C₂(½)²(½)⁵

1 - 1/128 - 7/128 - 21/128

1- 29/128 = 99/128

99/128 is not further reducible so, 99 and 128 are coprime.

99/128 = a/b, so a = 99 and b = 128

99 + 128 = 227

Answer 2

Pr(at least 3 of 7) = 1- Pr(0,1,2,)= over 50%

= 1 - Pr(0) - Pr(1) - Pr(2)

= 1- (1/2)^7 - 7(1/2)^7 - 21(1/2)^7

= 1- 29/128 = 3(3)(11)/2(2)(2)(2)(2)(2)(2)

= 99/128= a/b

a + b = 99+128 = 227

Pr( 3, 4, 5, 6 or 7) = 1- Pr(0,1, 2)