Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

A married couple decided to have $5$ children. Based on gene history, probability that any one of their children will need to wear eye glasses, independent of sex, is $60$%; probability that a child being a boy or a girl are equally $50$%. Let $X$ be the number of children that needs glasses and $Y$ be the number of boys in the family.

Probability distribution tables for $X$ and $Y$:

$$ \begin{array}{} \begin{array}{c|c|c} \text{X} & \text{P(X)}\\ \hline \\0 & 0.01024 \\1 & 0.07680 \\2 & 0.23040 \\3 & 0.34560 \\4 & 0.25920 \\5 & 0.07776 \end{array} & \begin{array}{c|c|c} \text{Y} & \text{P(Y)}\\ \hline \\0 & 0.03125 \\1 & 0.15625 \\2 & 0.31250 \\3 & 0.31250 \\4 & 0.15625 \\5 & 0.03125 \end{array} \end{array} $$

What is P(X=E(X))?

Let W be the number of girls that wear glasses. What is P(W=E(W))?

share|improve this question
2  
what do you calculate E(X) to be? –  oks Feb 12 '13 at 21:39
    
E(X) = 5 * 0.6 = 3 –  user60852 Feb 12 '13 at 21:45

1 Answer 1

up vote 1 down vote accepted

First part: $X\sim B(5,0.6)$, so $E(X)=np=3$ and $P(X=3)=0.34560$
Since being a girl and requiring glasses are independent, $W\sim B(5,p)$ where $p=0.5\cdot 0.6=0.3$. Thus, $E(W)=1.5$, and $P(W=1.5)=0$ since $W$ is discrete.

share|improve this answer
    
If P[W=E(W)] = 0 because W is discrete, why isn't P[E=E(X)] = 0? –  user60852 Feb 12 '13 at 22:07
1  
Because $3$ is an integer, but $1.5$ is not (I presume you mean $P(X=E(X))$). –  Daniel Littlewood Feb 12 '13 at 22:18
    
What is the probability that the number of children that wear glasses is equal the expected number of children that will wear glasses (3)? What is the probability that the number of girls that wear glasses is the expected number of girls that will wear glasses (1.5)? So if the above is the question both would be 0? –  user60852 Feb 12 '13 at 22:27
1  
The probability of $3$ children needing glasses is worked out above ($0.34560$). The other probability is $0$ because it is impossible to have one and a half girls that wear glasses. –  Daniel Littlewood Feb 12 '13 at 22:33
1  
@Daniel one might wear glasses and another a monocle –  Henry Feb 12 '13 at 23:43

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.