# If you flip a fair coin 4 times, what is the probability that you will get exactly 2 tails?

Mar 30, 2017

$P \left(\text{Exactly 2H}\right) = 0.375$

#### Explanation:

Method 1 - Tree Diagram

$P \left(\text{Exactly 2H") = P("HHTT") + P("HTHT}\right) +$
 " " P("HTTH") + P("TTHH") +
 " " P("THHT") + P("THTH")
$\text{ } = 0.0625 \cdot 6$
$\text{ } = 0.375$

Method 1 - Combinations

Using the combination formula:

 ""_nC^r = ( (n), (r) ) = (n!)/(r!(n-r)!)

We seek any combination of 2 heads from 4 coins:

 n("possible combinations") = ""_2C^4 = ( (4), (2) )
 " " = (4!)/(2!(4-2)!)
 " " = (4!)/(2!2!)
$\text{ } = \frac{24}{2 \cdot 2}$
$\text{ } = 6$

And the total number of all combinations of 4 flips

$n \left(\text{total combinations}\right) = {2}^{4}$
$\text{ } = 16$

$P \left(\text{Exactly 2H}\right) = \frac{6}{16}$
$\text{ } = 0.375$