A jar contains 2 red balls, 2 blue balls, 2 green balls and 1 orange ball. Balls are randomly selected, without replacement, until 2 of the same colour are obtained. Calculate the probability that more than 3 balls must be selected.
the answer is 7/15
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To calculate the probability that more than 3 balls must be selected to achieve 2 balls of the same color, we need to determine the total number of possible outcomes and the number of favorable outcomes.
First, let's consider the total number of possible outcomes. At most, we need to select 6 balls because there are only 7 balls in total. We can compute the total number of outcomes by summing the probabilities of selecting 4, 5, and 6 balls.
To calculate the probability of selecting 4 balls:
- Select any of the 4 red balls in the first selection (4/7)
- Select another red ball in the second selection (3/6)
- Since there are 2 red balls, we can have them arranged in two different ways: (red-red) or (red-red).
Therefore, we multiply by 2.
- Multiply all these probabilities together: (4/7) * (3/6) * 2
To calculate the probability of selecting 5 balls:
- Select any of the 4 red balls in the first selection (4/7)
- Select another red ball in the second selection (3/6)
- Select a ball of any other color in the third selection (3/5)
- Since there are 2 red balls, we can have them arranged in three different ways: (red-red-other), (red-other-red), or (other-red-red). Therefore, we multiply by 3.
- Multiply all these probabilities together: (4/7) * (3/6) * (3/5) * 3
To calculate the probability of selecting 6 balls:
- Select any of the 4 red balls in the first selection (4/7)
- Select another red ball in the second selection (3/6)
- Select a ball of any other color in the third selection (3/5)
- Select another ball of any other color in the fourth selection (2/4)
- Since there are 2 red balls, we can have them arranged in four different ways: (red-red-other-other), (red-other-red-other), (red-other-other-red), or (other-red-red-other). Therefore, we multiply by 4.
- Multiply all these probabilities together: (4/7) * (3/6) * (3/5) * (2/4) * 4
Now, let's calculate the probability of selecting more than 3 balls:
- Add up the probabilities of selecting 4, 5, and 6 balls: [(4/7) * (3/6) * 2] + [(4/7) * (3/6) * (3/5) * 3] + [(4/7) * (3/6) * (3/5) * (2/4) * 4]
The result is the probability that more than 3 balls must be selected to obtain 2 balls of the same color.