Suppose that exam scores were distributed normally. Let the mean be 80 and standard deviation be 8. If it is known that a student's score is greater than 75, what is the probability that his score is greater than 90?
I am a bit confused by the wording of the question. Should this be considered a case of conditional probability? Is my approach correct?
This is know I thought about it:
$$P(Y > 90 | Y >75) = \frac{P(Y > 90 \cap Y > 75)}{P(Y > 75)} = \frac{P(Y > 90)}{P(Y > 75)}.$$
Best Answer
Yes, It should definitely work. I am assuming that you know how to calculate probabilities of normal random variables. If that is the case then calculating the probability should be trivial.
Because the problem now reduces to $$\frac{P(Y\gt 90)}{P(Y\gt 75)}$$ I'll continue from thereon.
To compute $P(Y\gt 90)$ you first compute $P(Y\lt 90)$. To do his you first figure out how many standard deviations away from the mean is $90$. This is very often called the z-score, which you are aware of. So, $90$ is $10$ points away from your mean ($80$), this implies that 90 is $\frac{10}{8}=0.8 \sigma$s away from your mean. If you look up the probability of this particular z- score on a table, you'll see that $P(Y\lt90)=\Phi(0.8)=0.788.$ So
$P(Y\gt90)=1-\Phi(0.8)=0.212.$
Similarly, $P(Y\lt75)=\Phi(-0.625)=.266$. So $P(Y\gt75)=1-\Phi(-0.625)=.266=0.734$
So $$\frac{P(Y\gt 90)}{P(Y\gt 75)}=\frac{0.212}{0.734}=0.289$$
I hope It helps