Question

consider the following experiment: toss a coin twice and record the sequence of heads and tails.
a. what is the sample space (for tossing the coin twice)? select an answer
b. what is the event that you get at least one head? select an answer
c. what is the complement of the event that you get at least one head? select an answer
d. calculate the probability of the event in c.
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Explanation:

Step1: Recall Sample Space for Coin Toss

When tossing a coin twice, each toss has 2 outcomes (H for head, T for tail). The sample space \( S \) is all possible sequences: \( S = \{HH, HT, TH, TT\} \).

Step2: Define Event for "At Least One Head"

"At least one head" means 1 or 2 heads. So the event \( E \) is \( \{HH, HT, TH\} \).

Step3: Find Complement of \( E \)

The complement of an event \( E \), denoted \( E^c \), is all outcomes in \( S \) not in \( E \). So \( E^c = \{TT\} \) (no heads, i.e., two tails).

Step4: Calculate Probability of Complement

The probability of an event is the number of favorable outcomes over total outcomes. Total outcomes in \( S \): 4. Favorable for \( E^c \): 1. So \( P(E^c) = \frac{\text{Number of } E^c \text{ outcomes}}{\text{Total outcomes}} = \frac{1}{4} \).

Answer:

\(\frac{1}{4}\)