Question

1. A. From past experience it is known that 90% of one-year old children can distinguish their mother’s voice from the voice of a similar sounding female. A random sample of 50 one-year olds is given this voice recognition test.
   1. Find the probability that exactly 3 children do not recognize their mother’s voice. (2 points)

2. Find the probability that 3 or fewerchildren donot recognize their mother’s voice. (2 points)

3. What is the probability that at least 47 children recognize their mother’s voice?(2 points)

4. Find the probability that exactly 90% of the 50 children do recognize their mother’s voice. (2 points)

5. What is the probability that exactly 49 of the 50 children recognize their mother’s voice? (2 points)
   

6. Find the probability all 50 children recognize their mother’s voice. (2 points)

7. Let the random variable x denote the number of children who recognize their mother’s voice. Find the mean of x. (2 points)

8. Let the random variable x denote the number of children who do not recognize       their mother’s voice. Find the mean of x. (2 points)
   

1. 9. Let the random variable x denote the number of children who do not recognize their mother’s voice. Find the variance of x. (2 points)

Answer 1

p=0.1,n=45

$\dpi{100} a)P(k=3)=50C_3*0.1^3*0.9^{47}=0.1386$

$\dpi{100} b)P(k\leq 3)=\sum_{k=0}^350C_k*0.1^k*0.9^{50-k}=0.2503$

$\dpi{100} c)P(k'\geq 47)=P(k\leq 3)=\sum_{k=0}^350C_k*0.1^k*0.9^{50-k}=0.2503$

$\dpi{100} d)P(k=5)=50C_{5}*0.1^5*0.9^{45}=0.1849$

$\dpi{100} e)P(k=1)=50C_1*0.1^1*0.9^{49}=0.0286$

$\dpi{100} f)P(k=0)=50C_0*0.1^0*0.9^{50}=0.0052$

g)Mean=np=50*0.9=45 children recognize voice.

h)Mean=np=50*0.1=5 children donot recognize voice.

i)Variance=50*0.1*0.9=4.5