Question

7. The probability that a randomly chosen student will be left-handed is .09: a) In a class of 108, find the probability that there will be (exactly) 8 left-handed students. b) The classroom has 12 desks designed for people who are left-handed. Use an appropriate approximate method to find the probability that there are enough for all the left-handed students.

Answer 1

Solution :

Let x be student will be left handed

P = 0.09

Using the normal approximation to binomial distribution,

a). n = 108,

    μ = np = 9.72

    $\sigma = \sqrt{np(1-p)}$

       $=\sqrt{8.8452}$

       = 2.9741

Required probability,

P(x=8) = P(7.5 < x < 8.5)

$=P\left(\frac{7.5-9.72}{2.9741}< \frac{x-\mu}{\sigma}< \frac{8.5-9.72}{2.9741}\right)$

$=P(-0.75 < z < -0.41)$

Using z table,

= P(z<-0.41)-P(z<-0.75)

= 0.3409-0.2266

= 0.1143

b). P(x=12) = P(11.5 < x < 12.5)

$=P\left(\frac{11.5-9.72}{2.9741}< \frac{x-\mu}{\sigma}< \frac{12.5-9.72}{2.9741}\right)$

$=P(0.60 < z < 0.94)$

Using z table,

= P(z< 0.94)-P(z< 0.60)

= 0.8294 - 0.7257

= 0.1006