Use the rule for computing the probability of a union of two events?

If P(A∪B)=0.65, P(B)=0.40 and P(A)=0.75, find P(A∩B)

Solution :
P(A∪B) = P(A) + P(B) - P(A∩B)
P(A∩B) = P(A) + P(B) - P(A∪B)
P(A∩B) = 0.75 + 0.40 - 0.65
P(A∩B) = 0.50


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If a single card is drawn from a standard deck of 52 cards, what is the probability that it is either a ace or a red card?

Solution :
A : a ace
R : a red card

In a standard deck, there are 4 ace and 26 red cards (13 hearts and 13 diamond).
P(A) = 4/52
P(B) = 26/52

There are 2 red aces : 1 heart ace, 1 diamond ace
P(A∩B)
= P(a red ace)
= 2/52

P(a ace or a red card)
= P(A∪B)
= P(A) + P(B) - P(A∩B)
= (4/52) + (26/52) - (2/52)
= 28/52
= 7/13

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
0.65 = 0.75+0.40 - P(A ∩ B)

P(A ∩ B) = 0.75+0.40 -0.65 = 0.5

A = ace
R = red card

P(A) = 4/52 = 1/13
P(R) = 26/52 = 1/2
Ace and Red = 2/52 = 1/26
P(A ∪ R) = P(A) + P(R) - P(A ∩ R)
P(A ∪ R) = 1/13 + 1/2 - 1/26
= 7/13