Question

Three identical fair coins are thrown simultaneously until all three show the same face. What is the probability that they are thrown more than three times?

Answer 1

Answer:

The probability that the coins are thrown more than three times to show the same face is 0.3164.

Step-by-step explanation:

The problem is related to Geometric distribution.

The Geometric distribution defines the probability distribution of X failures before the first success.

The probability distribution function is:

$P(X=k)=(1-p)^{k}p;\ k = 0, 1, 2, ...$

First compute the probability that in the $i^{th}$ throw all the three coins will show the same face.

P (All the 3 coins shows the same face) = P (All the three coins shows Heads) + P (All the three coins shows Tails)

                      $=(\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2} )+(\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2} )\\=\frac{1}{8}+\frac{1}{8}\\ =\frac{2}{8} \\=\frac{1}{4}$

Now compute the probability that it takes more than 3 throws for the coins to show the same face.

P (X > 3) = 1 - P (X ≤ 3)

$=1-[P(X=1)+P(X=2)+P(X=3)]\\=1-[[(1-\frac{1}{4} )^{0}\times\frac{1}{4}]+[(1-\frac{1}{4} )^{1}\times\frac{1}{4}]+ [(1-\frac{1}{4} )^{2}\times\frac{1}{4}]+[(1-\frac{1}{4} )^{3}\times\frac{1}{4}]]\\=1-[0.2500+0.1875+0.1406+0.1055]\\=1-0.6836\\=0.3164$

Thus, the probability that it takes more than 3 throws for the coins to show the same face is 0.3164.