Question

These 3 problems:

1. A bag of marbles is filled with 8 green marbles, 5 blue marbles, 12 yellow marbles, and 10 red marbles. If two
marbles are blindly picked from the bag without replacement, what is the probability that exactly 1 marble will be
yellow?

2. A standard deck of cards contains 52 cards, 12 of which are called “face cards.” If the deck is shuffled and the
top two cards are revealed, what is the probability that at least 1 of them is a face card?

3. A delivery company has only an 8% probability of delivering a broken product when the item that is delivered is
not made of glass. If the item is made of glass, however, there is a 31% probability that the item will be delivered
broken. 19% of the company’s deliveries are of products made of glass. What is the overall probability of the
company delivering a broken product?

Answer 1

1. To find the probability that exactly 1 marble will be yellow, we need to consider the total number of marbles and the number of favorable outcomes.
Total number of marbles: 8 green + 5 blue + 12 yellow + 10 red = 35 marbles

Number of favorable outcomes: Choose 1 yellow marble out of 12 yellow marbles and choose 1 non-yellow marble out of the remaining 35 - 12 = 23 marbles.

Probability = (Number of favorable outcomes) / (Total number of outcomes)

Probability = (12/35) * (23/34) ≈ 0.281 or 28.1%

Therefore, the probability that exactly 1 marble will be yellow is approximately 28.1%.

2. The probability of at least 1 of the top two cards being a face card can be calculated by considering the complement probability (the probability of the opposite event, which is no face cards).
Total number of possible outcomes: Choosing 2 cards from a deck of 52 cards = 52C2 = (52 * 51) / (2 * 1) = 1326

Number of outcomes with no face cards: Choosing 2 non-face cards from the 40 non-face cards = 40C2 = (40 * 39) / (2 * 1) = 780

Number of outcomes with at least 1 face card = Total number of possible outcomes - Number of outcomes with no face cards = 1326 - 780 = 546

Probability = (Number of outcomes with at least 1 face card) / (Total number of outcomes)

Probability = 546/1326 ≈ 0.411 or 41.1%

Therefore, the probability that at least 1 of the top two cards is a face card is approximately 41.1%.

3. To find the overall probability of the company delivering a broken product, we need to consider the probability based on the type of item delivered (glass or non-glass) and the proportion of deliveries for each type.
Probability of delivering a broken product when the item is not made of glass = 8%

Probability of delivering a broken product when the item is made of glass = 31%

Proportion of deliveries for glass items = 19%

Proportion of deliveries for non-glass items = 1 - 19% = 81%

Overall probability = (Probability of delivering a broken non-glass item) * (Proportion of non-glass deliveries) + (Probability of delivering a broken glass item) * (Proportion of glass deliveries)

Overall probability = (0.08) * (0.81) + (0.31) * (0.19) ≈ 0.0808 + 0.0589 ≈ 0.1397 or 13.97%

Therefore, the overall probability of the company delivering a broken product is approximately 13.97%.

I hope this helps! :)