Question

Which of the following statements are true?

a. P(Box has a prize and Box is green) = P(Box has a prize) ✕ P(Box is green)
b. P(Box has a prize or Box is green) = P(Box has a prize) + P(Box is green)
c. P(Box has a prize | Box is green) = P(Box has a prize) ✕ P(Box is green)
d. P(Box has a prize and Box is green) = P(Box has a prize) + P(Box is green)

Answer 1

"P(Box has a prize | Box is green) = P(Box has a prize) ✕ P(Box is green)" is true. Thus the correct answer is option c. P(Box has a prize | Box is green) = P(Box has a prize) ✕ P(Box is green).

Step-by-step explanation:

The formula for conditional probability is given by:

`\[ P(A | B) = (P(A \cap B))/(P(B)) \]`

Let's apply this to the given statement:

`\[ P(\text Box is green) = \frac{P(\text{Box has a prize} \cap \text{Box is green})}{P(\text{Box is green})} \]`

Now, the given statement in option c is
`\( P(\textBox has a prize ) = P(\text{Box has a prize}) * P(\text{Box is green}) \),`

which aligns with the formula for conditional probability. Thus, option c is true.

Now, let's briefly explain the other options:

a. P(Box has a prize and Box is green) = P(Box has a prize) ✕ P(Box is green) is not true for general cases. It is only true if the events "Box has a prize" and "Box is green" are independent.

b. P(Box has a prize or Box is green) = P(Box has a prize) + P(Box is green) is not true. This is the formula for the probability of the union of two events, but it overcounts the case where both events happen.

d. P(Box has a prize and Box is green) = P(Box has a prize) + P(Box is green) is not a correct formulation for the probability of the intersection of two events.

In summary, option c is the correct statement based on the definition of conditional probability.