Question

the organized list show al the possible outcomes when 3 fair coins are flipped. heads = h, tails = t. what is the probability that at least 2 fair coins land heads up when 3 are flipped? (use a reduced fraction for your answer).

Explanation:

Step1: Identify total outcomes

The sample space has 8 outcomes (HHH, HHT, HTH, HTT, THH, THT, TTH, TTT). So total outcomes \( n = 8 \).

Step2: Identify favorable outcomes

"At least 2 heads" means 2 heads or 3 heads.

• Outcomes with 3 heads: HHH (1 outcome)
• Outcomes with 2 heads: HHT, HTH, THH (3 outcomes)

Total favorable outcomes \( m = 1 + 3 = 4 \).

Step3: Calculate probability

Probability \( P = \frac{m}{n} = \frac{4}{8} = \frac{1}{2} \)? Wait, no, wait: Wait, 1 (HHH) + 3 (HHT, HTH, THH) is 4? Wait, no, HHH is 3 heads, HHT, HTH, THH are 2 heads. So 1 + 3 = 4? Wait, but 4/8 reduces to 1/2? Wait, but let's check again. Wait, HHH (3H), HHT (2H), HTH (2H), THH (2H). So that's 1 + 3 = 4. Total outcomes 8. So 4/8 = 1/2? Wait, but the options have 1/2 as one of them. Wait, but wait, maybe I made a mistake. Wait, no: HHH (3H), HHT (2H), HTH (2H), THH (2H). So that's 4 outcomes. So 4/8 = 1/2. Wait, but let's check the sample space again. The sample space is:

HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. So:

• 3 heads: HHH (1)
• 2 heads: HHT, HTH, THH (3)
• 1 head: HTT, THT, TTH (3)
• 0 heads: TTT (1)

So "at least 2 heads" is 3 heads or 2 heads: 1 + 3 = 4. So probability is 4/8 = 1/2. Wait, but the option 1/2 is there. So the answer is 1/2.

Wait, but let's recheck. Wait, 4 favorable outcomes (HHH, HHT, HTH, THH) out of 8 total. So 4/8 = 1/2.

Answer:

\(\frac{1}{2}\) (The correct option is the one with 1/2, e.g., if the options are as given, the answer is the blue (wait, no, the options are 3/8, 6/8, 3/4, 1/2. So the correct option is the one with 1/2, i.e., the last option: 1/2).