Question

3 marbles are chosen at random without replacement from a bag containing 4 red marbles and 6 yellow marbles. Draw a tree diagram and use it to find:

(a) The probability that 2 red marbles are chosen.
(b) The probability of choosing at least two yellow marbles.

Answer 1

The probability of selecting 2 red marbles and 1 yellow marble is 10%, while the probability of selecting at least 2 yellow marbles is 45%, as calculated using a tree diagram for combinations without replacement.

Step-by-step explanation:

The probability of selecting marbles from a bag can be calculated using a tree diagram. In this scenario, there are three marbles chosen at random from a bag with 4 red and 6 yellow marbles, without replacement.

(a) The probability that 2 red marbles are chosen:

1. First draw: Probability of drawing a red marble is 4/10.
2. Second draw (after one red is already drawn): Probability of drawing another red marble is 3/9.
3. Third draw (regardless of what the second marble is): Probability of getting a non-red (yellow) marble is now 6/8.

Therefore, P(2 Red and 1 Yellow) = 4/10 x 3/9 x 6/8 = 0.10 or 10%.

(b) The probability of choosing at least two yellow marbles:

This situation can occur in three ways: YYR, YRY, or RYY.

1. For YYR: P(Yellow, Yellow, Red) = 6/10 x 5/9 x 4/8 = 0.15.
2. For YRY: P(Yellow, Red, Yellow) = 6/10 x 4/9 x 5/8 = 0.15.
3. For RYY: P(Red, Yellow, Yellow) = 4/10 x 6/9 x 5/8 = 0.15.

Add these probabilities together: 0.15 + 0.15 + 0.15 = 0.45 or 45%.