Probability of K People Tossing Coins n Times

[nbalderaz]

K people independently toss a coin n times. What is the probability that all K of them get the same number of heads?
(1) Find a formula for the probability in terms of K and n.
(2) Evaluate the probability for K = {5, 10, 20} and n = {4, 8, 16}

 

[Aphex_Twin]

Let me see if I remember this:


P(n)=C(n,x)*pi^x*(1-pi)^(n-x)

Where P(n) is the probability that an x number of events will occur out of n number of tries. pi is the certainty (in the case of a coin with 2 sides pi=1/2). C(n,x) is: n!/((n-x)!*k!)

Apply the formula K-1 times.

 

[tacman]

The probability of all K people getting the same number of heads when independently tossing a coin n times can be calculated by first determining the total number of possible outcomes, and then finding the number of outcomes where all K people get the same number of heads.

(1) Formula for the probability in terms of K and n:

Let P be the probability of all K people getting the same number of heads when tossing a coin n times.
Total number of possible outcomes = 2^n (since each person has 2 possible outcomes - heads or tails - for each toss)
Number of outcomes where all K people get the same number of heads = 2 (since there are only 2 possible outcomes - all heads or all tails)
Therefore, P = 2/2^n = 1/2^(n-1)

(2) Evaluating the probability for K = {5, 10, 20} and n = {4, 8, 16}:

For K = 5 and n = 4:
P = 1/2^(4-1) = 1/8 = 0.125 or 12.5%

For K = 5 and n = 8:
P = 1/2^(8-1) = 1/128 = 0.0078 or 0.78%

For K = 5 and n = 16:
P = 1/2^(16-1) = 1/32768 = 0.00003 or 0.003%

For K = 10 and n = 4:
P = 1/2^(4-1) = 1/8 = 0.125 or 12.5%

For K = 10 and n = 8:
P = 1/2^(8-1) = 1/128 = 0.0078 or 0.78%

For K = 10 and n = 16:
P = 1/2^(16-1) = 1/32768 = 0.00003 or 0.003%

For K = 20 and n = 4:
P = 1/2^(4-1) = 1/8 = 0.125 or 12.5%

For K = 20 and n = 8:
P = 1/2^(8-1) = 1/128 = 0.