A drawer contains 4 red socks and 5 blue. Three socks are drawn one at a time then put back before next selection. Determine the probability that:
i) exactly 2 red socks are selected.
ii) at least 2 red socks are selected
To determine the probability of selecting socks from the drawer, we need to calculate the number of favorable outcomes (sock combinations that meet the given conditions) and divide it by the total number of possible outcomes.
In this case, we have 4 red socks and 5 blue socks in the drawer.
i) To find the probability of exactly 2 red socks being selected:
Step 1: Calculate the total number of possible outcomes.
Since each sock is replaced after it is drawn, the total number of possible outcomes for each selection is equal to the total number of socks in the drawer:
Total possible outcomes = 4 (red socks) + 5 (blue socks) = 9 socks
Step 2: Calculate the number of favorable outcomes.
To select exactly 2 red socks, we need to:
Choose 2 red socks out of the 4 available: C(4, 2) = 6 (combination formula)
Choose 1 blue sock out of the 5 available: C(5, 1) = 5 (combination formula)
Multiply these two possibilities: 6 x 5 = 30
Step 3: Calculate the probability.
The probability of exactly 2 red socks being selected is the number of favorable outcomes divided by the total number of possible outcomes:
Probability = Favorable outcomes / Total possible outcomes
Probability = 30 / 9 ≈ 0.33 or 33.33%
ii) To find the probability of at least 2 red socks being selected:
Step 1: Calculate the total number of possible outcomes.
The total number of possible outcomes remains the same as in part i):
Total possible outcomes = 9 socks
Step 2: Calculate the number of favorable outcomes.
Here, we need to consider two cases:
Case 1: Selecting exactly 2 red socks, as calculated in part i) = 30 favorable outcomes.
Case 2: Selecting all 3 red socks, which is similar to selecting 3 red socks out of the 4 available: C(4, 3) = 4 favorable outcomes.
Add these two cases: 30 + 4 = 34 favorable outcomes.
Step 3: Calculate the probability.
The probability of at least 2 red socks being selected is the number of favorable outcomes divided by the total number of possible outcomes:
Probability = Favorable outcomes / Total possible outcomes
Probability = 34 / 9 ≈ 0.378 or 37.78%
Therefore:
i) The probability of exactly 2 red socks being selected is approximately 0.33 or 33.33%.
ii) The probability of at least 2 red socks being selected is approximately 0.378 or 37.78%.