Monique has a box that contains the following marbles: 5 yellow marbles, 3 green marbles, 2 orange marbles. She randomly selects 2 marbles from the box and keeps them. If she randomly selects a third marble from the box, the probability that she selects a green marble will be 2/10.
No, this statement is false. Let's calculate the probability of Monique selecting a green marble in her third draw after she has picked 2 marbles and kept them:
Total marbles in the box initially = 5 (yellow) + 3 (green) + 2 (orange) = 10
There are 3 possibilities for her first two selections:
1) She selects 2 yellow marbles (probability 5/10 * 4/9): The box will then contain 1 green and 2 orange marbles, and the probability of selecting a green marble will be 1/3.
2) She selects 1 yellow and 1 green marble or 1 green and 1 yellow marble (probability 5/10 * 3/9 + 3/10 * 5/9): The box will then contain 2 green and 2 orange marbles, and the probability of selecting a green marble will be 2/4 = 1/2.
3) She selects 2 green marbles (probability 3/10 * 2/9): The box will then contain 1 green and 2 orange marbles, and the probability of selecting a green marble will be 1/3.
Now let's calculate the overall probability of Monique selecting a green marble in her third draw using these probabilities:
Overall_probability = P(draw_green|GG) * P(GG) + P(draw_green|GY) * P(GY) + P(draw_green|YY) * P(YY)
Overall_probability = (1/3) * (3/10 * 2/9) + (1/2) * (5/10 * 3/9 + 3/10 * 5/9) + (1/3) * (5/10 * 4/9)
Overall_probability = 8/27 + 1/2 + 1/9 ≈ 0.481
This overall probability (48.1%) of Monique selecting a green marble in her third draw is different from 2/10 (20%); hence, the statement is false.
To find the probability that Monique selects a green marble on her third selection, we need to calculate the probability of the two cases where she already has a green marble and where she doesn't.
Case 1: Monique already has a green marble:
In this case, Monique has already selected one green marble, so there are 2 green marbles left in the box. There are also 7 marbles left in total (5 yellow marbles + 2 orange marbles). The probability of selecting a green marble on the third selection in this case is given by:
P(Green|Green) = (2 green marbles) / (7 total marbles) = 2/7
Case 2: Monique doesn't have a green marble:
In this case, Monique has not selected any green marbles yet, so there are still 3 green marbles in the box. There are 7 marbles left in total (5 yellow marbles + 2 orange marbles). The probability of selecting a green marble on the third selection in this case is given by:
P(Green|Not Green) = (3 green marbles) / (7 total marbles) = 3/7
To find the overall probability of selecting a green marble on the third selection, we need to consider both cases:
P(Green) = P(Green|Green) * P(Green) + P(Green|Not Green) * P(Not Green)
P(Green) = (2/7) * (3/10) + (3/7) * (7/10) = 6/70 + 21/70 = 27/70
Therefore, the probability that Monique selects a green marble on her third selection is 27/70.
To find the probability that Monique selects a green marble as her third marble, we need to determine the total number of possible outcomes and the number of favorable outcomes.
First, let's calculate the total number of possible outcomes. Monique selects 2 marbles first, and then she selects a third marble.
For the first selection, she has a total of 5 + 3 + 2 = 10 marbles to choose from.
For the second selection, she has a total of 9 marbles left since she already selected 2 marbles.
For the third selection, she has 8 marbles left since she already selected 2 marbles.
Therefore, the total number of possible outcomes is 10 * 9 * 8 = 720.
Next, let's calculate the number of favorable outcomes, which is the number of ways Monique can select a green marble as her third marble.
Monique initially has 3 green marbles, and she selects 2 marbles first. So there are 3C2 (3 choose 2) ways to select 2 green marbles.
After selecting 2 green marbles, she has only 1 green marble left for the third selection.
Therefore, the number of favorable outcomes is 3C2 * 1 = 3.
Finally, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes.
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 3 / 720
Probability = 1/240.
So the probability that Monique selects a green marble as her third marble is 1/240.