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.
"More than 3 balls" = either 4 or 5 balls.
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
Either-or probabilities are found by adding the individual probabilities.
To find the probability that more than 3 balls must be selected, we need to first calculate the total number of possible outcomes and then determine the number of favorable outcomes.
Total Number of Possible Outcomes:
The total number of balls in the jar is 2 red + 2 blue + 2 green + 1 orange = 7 balls. Since balls are selected without replacement, the total number of possible outcomes will be the number of ways to select any 2 balls from the 7 balls in the jar. This can be calculated using the combination formula.
The number of ways to select any 2 balls from 7 is given by C(7, 2) = 7! / (2! * (7-2)!) = 7! / (2! * 5!) = (7 * 6) / (2 * 1) = 21.
Therefore, the total number of possible outcomes is 21.
Favorable outcomes:
To calculate the number of favorable outcomes, we need to consider the different possibilities when identical colored balls are obtained in the first 3 selections:
1. Selecting 3 balls of the same color: This can happen in 4 ways (Red, Blue, Green, or Orange).
2. Selecting 2 balls of one color and 1 ball of another color: This can happen in 6 ways (RRB, RRG, RBB, RBG, GGB, GGR).
3. Selecting 2 balls of one color and 1 ball of a different color: This can happen in 6 ways (RRB, RRG, RBB, RGG, BBB, BGG).
Therefore, the total number of favorable outcomes is 4 + 6 + 6 = 16.
Probability:
To calculate the probability, we divide the number of favorable outcomes by the total number of possible outcomes:
Probability (more than 3 balls must be selected) = Favorable outcomes / Total outcomes = 16 / 21.
So the probability that more than 3 balls must be selected is 16/21, which can also be expressed as approximately 0.762 or 76.2%.