An urn contains 4 red and 3 blue marbles. What is the probability when three marbles are selected randomly that all 3 will be red, if we select each marble
a) with replacement
b) without replacement
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with replacement:
prob = (4/7)(4/7)(4/7)
without replacement:
prob = (4/7)(3/6)(2/5)
To calculate the probability, we need to consider both the total number of outcomes and the favorable outcomes.
a) With Replacement:
When selecting with replacement, after each marble is selected, it is put back into the urn before the next selection. This means that the total number of marbles in the urn remains the same for each selection.
Total number of outcomes = number of marbles^number of selections = 7^3 = 343
Favorable outcomes = number of red marbles^number of selections = 4^3 = 64
Therefore, the probability of selecting all 3 marbles to be red with replacement is 64/343 ≈ 0.187.
b) Without Replacement:
When selecting without replacement, each marble is not put back into the urn before the next selection. This decreases the total number of marbles in the urn for each subsequent selection.
Total number of outcomes = number of marbles * (number of marbles - 1) * (number of marbles - 2) = 7 * 6 * 5 = 210
Favorable outcomes = number of red marbles * (number of red marbles - 1) * (number of red marbles - 2) = 4 * 3 * 2 = 24
Therefore, the probability of selecting all 3 marbles to be red without replacement is 24/210 ≈ 0.114.