Question

we have a deck of 10 cards numbered from 1 to 10. some are grey and some are white, as shown below.
image of cards: 1 (grey), 2 (white), 3 (white), 4 (white), 5 (grey); 6 (white), 7 (white), 8 (white), 9 (grey), 10 (grey)
the cards numbered 1, 5, 9, and 10 are grey. the cards numbered 2, 3, 4, 6, 7, and 8 are white.
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 white?
(b) what is the probability that the card drawn is white, given that the number drawn is less than 9?

Explanation:

Part (a)

Step1: Identify total and white cards

Total cards: \( n = 10 \). White cards: 2, 3, 4, 6, 7, 8 → \( 6 \) cards.

Step2: Calculate probability

Probability \( P(\text{white}) = \frac{\text{Number of white cards}}{\text{Total number of cards}} = \frac{6}{10} = \frac{3}{5} \).

Step1: Identify cards < 9

Numbers less than 9: 1, 2, 3, 4, 5, 6, 7, 8 → \( 8 \) cards.

Step2: Identify white cards < 9

White cards in these: 2, 3, 4, 6, 7, 8 → \( 6 \) cards.

Step3: Calculate conditional probability

\( P(\text{white} | \text{number} < 9) = \frac{\text{White cards < 9}}{\text{Cards < 9}} = \frac{6}{8} = \frac{3}{4} \).

Answer:

\(\frac{3}{5}\)

Part (b)