Getting Two Heads in Four Tosses of a Coin

Date: 05/17/2000 at 22:01:23
From: Melissa 
Subject: Probability of two heads on four tosses

Dear Dr. Math,

The question I need to ask is: What is the probability of getting two 
heads on four flips of an unbiased coin? I have looked at your other 
answers, and think it would be 1/8 because:

     1/2 + 1/2 + 1/2 + 1/2 = 1/8

Thanks so much for your help.

Melissa Dismukes

Date: 05/18/2000 at 13:42:43
From: Doctor TWE
Subject: Re: Probability of two heads on four tosses

Hi Melissa - thanks for writing to Dr. Math.

I think what you meant was (1/2)*(1/2)*(1/2)*(1/2) = 1/16, but that 
would be the answer for getting 4 heads in 4 flips (or 0 heads in 4 
flips).

Let's draw a probability tree:


                                 /\                               
Toss: 
                              /       \
                          /               \
                      /                       \
                  T                               H                1st
                  |                               |
                 / \                             / \
               /     \                         /     \
            /           \                   /           \
          T               H               T               H        2nd
          |               |               |               |
         / \             / \             / \             / \
       /     \         /     \         /     \         /     \
      T       H       T       H       T       H       T       H    3rd
      |       |       |       |       |       |       |       |
     / \     / \     / \     / \     / \     / \     / \     / \
    T   H   T   H   T   H   T   H   T   H   T   H   T   H   T   H  4th
    |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
   0H  1H  1H  2H  1H  2H  2H  3H  1H  2H  2H  3H  2H  3H  3H  4H
   4T  3T  3T  2T  3T  2T  2T  1T  3T  2T  2T  1T  2T  1T  1T  0T
                *       *   *           *   *       *

Since it is an unbiased coin, each branch has a probability of 1/2, 
each outcome is equiprobable (P = 1/16), and the probability of 
tossing exactly 2 heads (outcomes marked with *) is 6/16 = 3/8.

For a more general solution to this type of problem, search Dr. Math 
for "binomial probability" (without the quotation marks) using our 
archive search engine at:

   http://mathforum.org/mathgrepform.html   

We have many archived questions and answers on this topic.

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/