A coin is thrown 3 times. What is the probability that atleast one head is obtained?

1. Analyze the question

We are tossing a fair coin 3 times. We need to find the probability of getting at least one head.

2. Explain the concept

When dealing with "at least one" problems in probability, it is often much faster to use the Complement Rule. The sum of the probability of an event occurring and the probability of it not occurring is always 1.

• $P(\text{at least one head}) = 1 - P(\text{no heads})$

"No heads" is the same as getting "all tails" (TTT).

3. Show the solution

• Step 1: Find the total number of possible outcomes. Since a coin has 2 sides (Head or Tail) and it is tossed 3 times, the total outcomes = $2^3 = 2 \times 2 \times 2 = 8$. The sample space is: {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}.

• Step 2: Identify the outcome for "no heads". The only way to get no heads is to get tails on all three tosses: {TTT}. There is only 1 such outcome.

• Step 3: Calculate the probability of no heads. $P(\text{no heads}) = \frac{\text{Number of "no head" outcomes}}{\text{Total outcomes}} = \frac{1}{8}$

• Step 4: Apply the complement rule. $P(\text{at least one head}) = 1 - P(\text{no heads})$ $P(\text{at least one head}) = 1 - \frac{1}{8}$ $P(\text{at least one head}) = \frac{8}{8} - \frac{1}{8} = \frac{7}{8}$

4. Identify the answer

Comparing our result to the options provided, the correct choice is A.

5. Exam Tip

Whenever you see the phrase "at least one" in a probability question, immediately think: "1 minus the probability of none." It saves you from having to list out all the combinations for 1 head, 2 heads, and 3 heads separately!