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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
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/
```
Associated Topics:
High School Probability
Middle School Probability

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