Question

we have a deck of 8 cards numbered from 1 to 8. some are white and some are grey, as shown below. the cards numbered 1, 4, 6, and 8 are white. the cards numbered 2, 3, 5, and 7 are grey. answer the following questions. write each answer as a fraction. a card will be drawn at random. (a) what is the probability that the card drawn is grey? (b) what is the probability that the card drawn is grey, given that an even - numbered card is drawn?

Explanation:

Step1: Calculate probability for part (a)

Probability formula: \(P(A)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\).
Total cards \(n = 8\). Grey cards \(m=4\) (cards \(2\), \(3\), \(5\), \(7\)).
So \(P(\text{grey})=\frac{4}{8}\).

Step2: Simplify the fraction for part (a)

Simplify \(\frac{4}{8}\) by dividing numerator and denominator by \(4\). \(\frac{4\div4}{8\div4}=\frac{1}{2}\).

Step3: Calculate probability for part (b)

Even - numbered cards: \(2\), \(4\), \(6\), \(8\) (\(n_{even}=4\)).
Grey and even - numbered card: \(2\) (\(m_{grey\cap even}=1\)).
Using conditional probability formula \(P(A|B)=\frac{P(A\cap B)}{P(B)}\), here \(P(\text{grey}|\text{even})=\frac{\text{Number of grey and even cards}}{\text{Number of even cards}}\).
So \(P(\text{grey}|\text{even})=\frac{1}{4}\).

Answer:

(a) \(\frac{1}{2}\)
(b) \(\frac{1}{4}\)