Given the following probabilities, for what P(B)
are events A
and B
independent?
P(A)=34
P(A and B)=14
Responses
23
2 thirds
13
1 third
38
3 eighths
12
The events A and B are independent if P(A and B) = P(A) * P(B).
From the given probabilities, we have P(A) = 3/4 and P(A and B) = 1/4.
For events A and B to be independent, we need to find P(B) such that P(A and B) = P(A) * P(B).
Therefore,
1/4 = (3/4) * P(B)
Multiplying both sides of the equation by 4, we have:
1 = 3 * P(B)
Dividing both sides of the equation by 3, we find:
1/3 = P(B)
So, for events A and B to be independent, P(B) must be 1/3.
An experiment consists of drawing a single card from a standard deck of 52
cards. Event A
is "drawing a heart" and event B
is "drawing a face card (jack, queen, or king)."
Are events A
and B
dependent or independent? Why?
Select the option that correctly answers both questions.
Responses
Events A
and B
are dependent, because P(A)≠P(B|A).
Events cap A and cap b are dependent, because P(A)≠P(B|A).
Events A
and B
are independent, because P(A)=P(B|A)=14.
Events cap A and cap b are independent, because
Events A
and B
are dependent, because P(B)≠P(B|A).
Events cap A and cap b are dependent, because P(B)≠P(B|A).
Events A
and B
are independent, because P(B)=P(B|A)=313.
Events A and B are independent, because P(A) = P(B|A).
In this case, P(A) is the probability of drawing a heart, which is 13/52 (there are 13 hearts in a deck of 52 cards).
P(B|A) is the probability of drawing a face card given that we have already drawn a heart. Since there are 3 face cards (jack, queen, and king) in a deck of 52 cards, and 1 of them (the king of hearts) is already accounted for in event A, the probability of drawing a face card given that we have already drawn a heart is 2/51.
Therefore, since P(A) = P(B|A) = (13/52) = (2/51), events A and B are independent.
To determine if events A and B are independent, we need to check if the probability of A occurring is the same regardless of the occurrence of B.
If events A and B are independent, then P(A and B) = P(A) * P(B).
Given that P(A) = 34 and P(A and B) = 14, let's substitute these values into the equation:
14 = 34 * P(B)
To solve for P(B), divide both sides of the equation by 34:
14/34 = P(B)
Simplifying the fraction, we get:
7/17 = P(B)
Therefore, events A and B are independent if P(B) = 7/17.