Question

Problem from statistic.

1.Problem

A research project analyzes cognitive abilities of left-handed children. In a small scale pre-study, 6 children are selected randomly from a kindergarten group of 20 children. Assume that 10% of the children in this kindergarten group are left-handers.

1. Find the probability that the sample of 6 children includes exactly one left-handed child. Note: Define appro- priate random variables and their distributions.
2. Find the probability that the sample includes more than two left-handed children.

2.Problem

An internet mail order business sends out emails to current clients to advertise a new product. It is known from previous advertising campaigns that the probability of an order is 0.18. The number of current clients is n = 100, 000. The product price is 70 euros.

1. Calculate the expected sales for the campaign. Note: define appropriate random variables assuming indepen- dence of orders.
2. Calculate the probability that sales are greater than 1,274,980 euros.

Please use Cumulative Areas of the Standard Normal Distribution

Answer 1

1)

Let , X be the number of left-handed children.

Here , X has binomial distribution with parameter n=6 ans p=0.10

The probability mass function of X is ,

; x=0,1,2,....,n and q=1-p

= 0 ; otherwise

Now the probability distribution table is ,

X				P(X=x)
0	1	1	0.531441	0.5314
1	6	0.1	0.59049	0.3543
2	15	0.01	0.6561	0.0984
3	20	0.001	0.729	0.0146
4	15	0.0001	0.81	0.0012
5	6	0.00001	0.9	5E-05
6	1	1E-06	1	1E-06

Now ,

1)

Therefore , the probability that the sample of 6 children includes exactly one left-handed child is 0.3543.

2)

=1-(0.0984+0.3543+0.5314)

=0.0159

Therefore , the probability that the sample includes more than two left-handed children is 0.0159