Question

Lucy is taking English and history. Let E be the event that Lucy gets an A in her English class and let H be the event that she gets an A in her history class. At the beginning of the term these probabilities were P(E) = 0.60 P(H) = .70 P(E and H) = 0.55

(a) Are the events E and H independent? Explain how you know.

(b) Find the probability that Lucy will get at least one A between her English and history classes

Answer 1

P(E)=0.60

P(H)=0.70

P(E$\cap$H)=0.55

(a) For the events E and H to be independent there is at least one of the below condition must be true

• P(E|H)=P(E)
• P(H|E)=P(H)
• P(E∩H)=P(E)P(H)

P(E|H)=$\dfrac{P(E\cap H)}{H}=\dfrac{0.55}{0.70}=0.785\neq P(E)$

P(H|E)=$\dfrac{P(H\cap E)}{E}=\dfrac{0.55}{0.60}=0.91\neq P(H)$

P(E).P(H)=0.60$\times$ 0.70=0.42$\neq P(E\cap H)$

Hence, E and H are dependent events.

(b) Probability that Lucy will get at least one A between her English and history class is $P(E\cup H)$

So, By formula $P(E\cup H)=P(E)+P(H)-P(E\cap H)$

$=0.60+0.70-0.55=0.75$

Hence, the probability that Lucy will get at least one A between her English and history class is 0.75