Photo AI

The events A and B are such that P(A) = 0.7, P(B) = 0.5 and P(A ∩ B) = 0.3 - Leaving Cert Mathematics - Question 1 - 2012

Question 1

The events A and B are such that P(A) = 0.7, P(B) = 0.5 and P(A ∩ B) = 0.3. (a) Find P(A ∪ B) (b) Find P(A | B) (c) State whether A and B are independent events, ... show full transcript

Worked Solution & Example Answer:The events A and B are such that P(A) = 0.7, P(B) = 0.5 and P(A ∩ B) = 0.3 - Leaving Cert Mathematics - Question 1 - 2012

Step 1

Find P(A ∪ B)

96%

114 rated

Answer

To find the probability of the union of two events, we use the formula:

$P(A ∪ B) = P(A) + P(B) - P(A ∩ B)$

Substituting the given values:

$P(A ∪ B) = 0.7 + 0.5 - 0.3 = 0.9$

Thus, the probability of A or B occurring is 0.9.

Step 2

Find P(A | B)

99%

104 rated

Answer

To find the conditional probability of A given B, we use the formula:

$P(A | B) = \frac{P(A ∩ B)}{P(B)}$

Substituting the known values:

$P(A | B) = \frac{0.3}{0.5} = 0.6$

Hence, the conditional probability P(A | B) is 0.6.

Step 3

State whether A and B are independent events, and justify your answer.

96%

101 rated

Answer

Two events A and B are considered independent if:

$P(A ∩ B) = P(A) \cdot P(B)$

Calculating the right side:

$P(A) \cdot P(B) = 0.7 \cdot 0.5 = 0.35$

Given that P(A ∩ B) = 0.3, we can see that:

$0.3 \neq 0.35$

Thus, A and B are not independent events, as their intersection probability does not equal the product of their individual probabilities.

97% of Students

Report Improved Results

98% of Students

Recommend to friends

100,000+

Students Supported

1 Million+

Questions answered

The events A and B are such that P(A) = 0.7, P(B) = 0.5 and P(A ∩ B) = 0.3 - Leaving Cert Mathematics - Question 1 - 2012

Worked Solution & Example Answer:The events A and B are such that P(A) = 0.7, P(B) = 0.5 and P(A ∩ B) = 0.3 - Leaving Cert Mathematics - Question 1 - 2012

Find P(A ∪ B)

Find P(A | B)

State whether A and B are independent events, and justify your answer.

Join the Leaving Cert students using SimpleStudy...

Other Leaving Cert Mathematics topics to explore