Question

In a county containing a large number of rural homes, 60% of the homes are insured against fire. Four rural homeowners are chosen at random from this county, and x are found to be insured against fire. Find the probability distribution for x.

Answer 1

`\begin{array}{cccccc}x & {0} & {1} & {2} & {3} & {4} \ \\ P(x) & {0.0256} & {0.1536} & {0.3456} & {0.3456} & {0.1296} \ \end{array}`

Explanation:

Given

`p = 60\%`

`n = 4`

Required

The distribution of x

The above is an illustration of binomial theorem where:

`P(x) = ^nC_x * p^x *(1 - p)^(n-x)`

This gives:

`P(x) = ^4C_x * (60\%)^x *(1 - 60\%)^(n-x)`

Express percentage as decimal

`P(x) = ^4C_x * (0.60)^x *(1 - 0.60)^(n-x)`

`P(x) = ^4C_x * (0.60)^x *(0.40)^(4-x)`

When x = 0, we have:

`P(x=0) = ^4C_0 * (0.60)^0 *(0.40)^(4-0)`

`P(x=0) = 1 * 1 *(0.40)^4`

`P(x=0) = 0.0256`

When x = 1

`P(x=1) = ^4C_1 * (0.60)^1 *(0.40)^(4-1)`

`P(x=1) = 4 * (0.60) *(0.40)^3`

`P(x=1) = 0.1536`

When x = 2

`P(x=2) = ^4C_2 * (0.60)^2 *(0.40)^(4-2)`

`P(x=2) = 6 * (0.60)^2 *(0.40)^2`

`P(x=2) = 0.3456`

When x = 3

`P(x=3) = ^4C_3 * (0.60)^3 *(0.40)^(4-3)`

`P(x=3) = 4 * (0.60)^3 *(0.40)`

`P(x=3) = 0.3456`

When x = 4

`P(x=4) = ^4C_4 * (0.60)^4 *(0.40)^(4-4)`

`P(x=4) = 1 * (0.60)^4 *(0.40)^0`

`P(x=4) = 0.1296`

So, the probability distribution is:

`\begin{array}{cccccc}x & {0} & {1} & {2} & {3} & {4} \ \\ P(x) & {0.0256} & {0.1536} & {0.3456} & {0.3456} & {0.1296} \ \end{array}`