Chapter 6.2, Problem 38E

Solution

To make: A histogram for the given binomial experiment and find the most likely number of success. Also describe the distribution as symmetric or skewed.

  

Most likely number of success is 6.

Shape of the distribution is symmetric.

Given information:

Number of trials, and probability of success is given as follows:

  $n=12$ , $p=0.5$

Formula used:

Formula of Binomial distribution of $n$ trail of an event with probability of success

  $p$ for exactly $k$ success is

  $P\left(X=k\right)=C{}_k{\left(p\right)}^k{\left(1-p\right)}^{n-k}$

The most likely sum will be the $k$ , which has greatest $P\left(k\right)$ .

Shape of the distribution will be symmetric if left half of the distribution is the mirror image of right half.

Shape of the distribution will be skewed if the distribution is not symmetric about any vertical line.

Calculation:

Use the formula $P\left(X=k\right)=C{}_k{\left(p\right)}^k{\left(1-p\right)}^{n-k}$ to make a table for the binomial experiment.

$k$	Binomial distribution
0	2.44140625E-4
1	0.0029296875
2	0.01611328125
3	0.0537109375
4	0.1208496094
5	0.193359375
6	0.2255859375
7	0.193359375
8	0.1208496094
9	0.0537109375
10	0.01611328125
11	0.0029296875
12	2.44140625E-4

Draw the histogram for the binomial experiment as shown:

  

Probability corresponding to 6 success is greatest so the most likely number of success is 6.

Shape of the distribution will symmetric since the left half of the distribution is mirror image of the right half about a vertical line.