From past experience, it is known 90% of one-year-old children can distinguish their mother's voice from the voice of a similar sounding female. A random sample of 20 one-year-olds is given this voice recognition test. a) Find the probability at least 3 children do not recognize their mother's voice. P(X ≥ 3) = 0.323 (Here P = 0.10)

Step-by-step explanation:

Hello!

The study variable is

X: number of one-year-old children that cannot distinguish their mother's voice from the voice of a similar-sounding female, from a sample of 20.

Be this a binomial experiment where the "success" is that the kid does not distinguish his mother's voice and "failure" when the kid distinguishes his mother's voice.

Binomial criteria:

1. The number of observation of the trial is fixed n = 20

2. Each observation in the trial is independent, this means that none of the trials will affect the probability of the next trial. Each one-year-old is independent of the rest.

3. The probability of success in the same from one trial to another.

If we know that the probability of the kid distinguishing his mother, i.e. the probability of "failure", is 0.9, then the probability of the kid not distinguishing it is 1 - 0.9= 0.10. For this binomial experiment, the probability of success is p= 0.10

So X≈ Bi (n;ρ) with n=20 and p=0.1

P(X≥3)

This expression includes all possible outcomes from 3 to 20. You have several ways to resolve this.

1) Using you can calculate the point probabilities from 3 to 20 and then add them.

2) If the expression includes all values from 3 to 20 then it leaves out X=0, X=1, and X=2, so you can rewrite it as:

P(X≥3) = 1 - P(X≤2)

Then use the formula to calculate the point probabilities for 0, 1 and 2, add them and then subtract them from one.

Or, a more simple way is to use a table of the binomial distribution and get from it the already calculated value of P(X≤2).

I'll use the table n=20;p=0.10;x=2 ⇒ P(X≤2)= 0.6769

Then:

P(X≥3) = 1 - P(X≤2)= 1 - 0.6769= 0.3231

I hope it helps!