suppose you have 10 green marbles, 10 gold marbles, and ten black marbles. You also have 3 bags. Your plan is to ask a friend to randomly select a marble from one bag. How should you distribute all the marbles amongthe 3 bags to maximize the probability that your friend will choose a green marble? calculate that maximum probability.
To maximize the probability of your friend choosing a green marble, you should distribute the marbles in the following way:
1. Place all the 10 green marbles in one bag.
2. Put 9 gold marbles and 1 black marble in another bag.
3. Place all the remaining marbles (10 gold marbles, 9 black marbles) in the third bag.
By distributing the marbles in this manner, you ensure that the bag with the highest proportion of green marbles (1 green marble out of 10 total marbles) is selected with the highest probability.
To calculate the maximum probability, we need to consider the distribution method described above.
Assuming your friend randomly chooses a bag, there is a 1/3 probability of selecting any particular bag.
The probability of selecting a green marble will be different depending on the bag chosen:
- Bag 1 (10 green marbles and 0 other marbles): The probability of selecting a green marble = (Number of green marbles in the bag) / (Number of total marbles in the bag) = 10/10 = 1.
- Bag 2 (1 green marble, 9 gold marbles, and 1 black marble): The probability of selecting a green marble = 1/11.
- Bag 3 (0 green marbles, 10 gold marbles, and 9 black marbles): The probability of selecting a green marble = 0/19 = 0.
Since there is an equal chance of selecting each bag, we need to calculate the weighted average of these probabilities.
Weighted Average = (Probability of selecting Bag 1) * (Probability of selecting a green marble in Bag 1) + (Probability of selecting Bag 2) * (Probability of selecting a green marble in Bag 2) + (Probability of selecting Bag 3) * (Probability of selecting a green marble in Bag 3).
Weighted Average = (1/3) * 1 + (1/3) * (1/11) + (1/3) * 0 = 1/3 + 1/33 = 11/33 ≈ 0.3333.
Therefore, the maximum probability that your friend will choose a green marble is approximately 0.3333 or 33.33%.