To find the probability that at least 3 of the balls drawn are red when Malee randomly picks 8 balls from a bag containing 9 red, 4 green, and 3 yellow balls, we can solve the problem using the concept of combinations and complementary counting.

Step 1: Total number of balls in the bag

• Total number of balls = $9 + 4 + 3 = 16$ balls.

Step 2: Total possible outcomes

Malee picks 8 balls from 16 without replacement. The total number of ways to do this is the combination:

$\binom{16}{8} = \frac{16!}{8!(16-8)!} = 12870$

Step 3: Complementary counting (less than 3 red balls)

We can calculate the number of outcomes where Malee draws fewer than 3 red balls (0, 1, or 2 red balls) and subtract that from the total to get the number of favorable outcomes (i.e., at least 3 red balls).

Case 1: 0 red balls

Malee selects all 8 balls from the 7 non-red balls (4 green + 3 yellow). The number of ways to do this is:

$\binom{7}{8} = 0 \quad \text{(not possible as there are only 7 non-red balls)}$

Case 2: 1 red ball

Malee selects 1 red ball and 7 non-red balls. The number of ways to do this is:

$\binom{9}{1} \times \binom{7}{7} = 9 \times 1 = 9$

Case 3: 2 red balls

Malee selects 2 red balls and 6 non-red balls. The number of ways to do this is:

$\binom{9}{2} \times \binom{7}{6} = 36 \times 7 = 252$

Step 4: Total unfavorable outcomes

The total number of ways to select fewer than 3 red balls is the sum of all the unfavorable cases:

$0 + 9 + 252 = 261$

Step 5: Number of favorable outcomes

The number of favorable outcomes (at least 3 red balls) is the complement of the unfavorable outcomes:

$12870 - 261 = 12609$

Step 6: Probability of at least 3 red balls

Finally, the probability that Malee draws at least 3 red balls is the ratio of favorable outcomes to the total possible outcomes:

$P(\text{at least 3 red}) = \frac{12609}{12870} \approx 0.9797$

Thus, the probability that at least 3 of the balls drawn are red is approximately $0.9797$ or 97.97%.

Would you like more details or have any questions?

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Here are five related questions for further practice:

1. What is the probability that exactly 3 of the balls drawn are red?
2. If Malee draws 5 balls instead of 8, what is the probability that at least 2 of them are red?
3. What is the probability that none of the balls drawn are green?
4. What is the probability that all of the balls drawn are of the same color?
5. If Malee draws 6 balls, what is the probability that she gets exactly 4 red balls?

Tip: When calculating probabilities for complex events, complementary counting can simplify the problem significantly by focusing on what we don't want first!