Key to "What is the probability of each of the..."

ambiguit

Answered question

2021-11-14

What is the probability of each of the following events:
1) What is the probability that a fair coin lands heads exactly 15 times out of 18 flips?
2) If a biased coin has a probability of heads that is twice the probability of tails and is flipped 100 times, what is the probability of exactly ten heads?

Answer & Explanation

mylouscrapza

Beginner2021-11-15Added 22 answers

Step 1
1) When we flip a coin then the outcomes are head and tail.
So the probability of getting a head $= \frac{1}{2}$
number of observation $= 18$
Let $X =$ number of heads
So, X follows binomial distribution with $n = 18 and p = \frac{1}{2}$
We have to find $P (X = 15)$
$P (X = 15)$
$= (\begin{matrix} 18 \\ 15 \end{matrix}) (\frac{1}{2})^{15} (1 - \frac{1}{2})^{18 - 15}$
$= \frac{18!}{15! 3!} {(\frac{1}{2})}^{15} {(\frac{1}{2})}^{3}$
$= 816 {(\frac{1}{2})}^{18} = \frac{816}{2^{18}} = \frac{51}{16384}$
$= 0.00311279296875$
Answer: $\frac{51}{16384} or 0.00311279296875$
Step 2
2) Let the probability of tails $= x$
So the probability of head $= 2 x$
Total probability $= 1$
So, $x + 2 x = 1$
$3 x = 1$
$x = \frac{1}{3}$
$2 x = \frac{2}{3}$
number of observations $= n = 100$
Let, $X =$ number of heads
So, X follows binomial distribution with $n = 100 and p = \frac{2}{3}$
We have to find $P (X = 10)$
$P (X = 10)$
$= b \in o \mp d f (100, \frac{2}{3}, 10)$
$= (\begin{matrix} 100 \\ 10 \end{matrix}) (\frac{2}{3})^{10} (1 - \frac{2}{3})^{100 - 10}$
$= \frac{100!}{10! 90!} {(\frac{2}{3})}^{10} {(\frac{1}{3})}^{90}$
$= 3.439373308 \times 10^{- 32}$
Answer: $3.439373308 \times 10^{- 32}$

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