Question

When Ayla plays darts the chances that she hits bulls eye is 0.5. (Bulls Eye is the centre of the dart board). Answer the questions below:1. What are the chances that three darts fired in succession will all hit bulls eys? Round answer to three decimal places.2. What is the probability that none will hit? Round answer to three decimal places.3. What is the probability that a least one dart will hit? Round answer to three decimal places.

Answer 1

Given that,

When Ayla plays darts the chances that she hits bulls eye is 0.5.

To find: The chances that three darts fired in succession will all hit bulls eye.

Consider p be the probability of success.

Here,

p=0.5

q be the probability of failure,

`q=1-p=1-0.5=0.5`

By binomial distribution, we have that,

`P(X=x)=nC_xp^xq^(n-x)`

where n is the number of times the event is repeated and x is the number of favorable outcomes.

Here, n=3, we get

`P(X=x)=3C_xp^xq^(3-x)`

Here, x=3, we get,

`P(X=3)=3C_3p^3q^0`
`=(0.5)^3=0.125`

The required probability is 0.125.

2)To find the probability that none will hit.

Here, x=0, substitute the value in P(X=x), we get

`P(X=0)=3C_0p^0q^3`
`=(0.5)^3=0.125`

The required probability is 0.125.

3) To find the probability that a least one dart will hit.

Here, to find P(X>0), we get,

`P(X>0)=P(X=1)+P(X=2)+P(X=3)`

Substitute the values we get,

`=3C_1(0.5)(0.5)^2+3C_2(0.5)^2(0.5)+(0.5)^3`
`=3(0.5^)^3+3(0.5)^3+(0.5)^3`
`=0.75+0.125`
`=0.875`

The required probability is 0.875