A coin is tossed three times. Find the probability of the following events.

(1) A: getting at least two heads

(2) B: getting exactly two heads.

(3) C: getting at most one head.

To determine the probabilities of the events when a coin is tossed three times, we first need to understand the total number of possible outcomes. Each toss of a coin has two possible results: heads (H) or tails (T). When tossing the coin three times, the total number of outcomes can be calculated using the formula for combinations of independent events.

Total Outcomes

For three tosses, the total number of outcomes is calculated as follows:

• Each toss has 2 outcomes (H or T).
• For three tosses: 2 × 2 × 2 = 2³ = 8.

Thus, the possible outcomes are: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

Event A: Getting at least two heads

To find the probability of getting at least two heads, we need to identify the outcomes that meet this criterion. The outcomes with at least two heads are:

• HHH
• HHT
• HTH
• THH

Counting these, we find there are 4 favorable outcomes. Therefore, the probability of event A is:

P(A) = Number of favorable outcomes / Total outcomes = 4 / 8 = 1/2.

Event B: Getting exactly two heads

Next, we look for the outcomes that result in exactly two heads. The favorable outcomes for this event are:

• HHT
• HTH
• THH

There are 3 outcomes that satisfy this condition. Thus, the probability of event B is:

P(B) = Number of favorable outcomes / Total outcomes = 3 / 8.

Event C: Getting at most one head

For event C, we need to find the outcomes that have at most one head. The outcomes that fit this description are:

• TTT
• TTH
• THT
• HTT

Counting these, we see there are 4 outcomes. Therefore, the probability of event C is:

P(C) = Number of favorable outcomes / Total outcomes = 4 / 8 = 1/2.

Summary of Probabilities

To summarize, the probabilities for the events are as follows:

• P(A) = 1/2 (at least two heads)
• P(B) = 3/8 (exactly two heads)
• P(C) = 1/2 (at most one head)

Understanding these probabilities helps in grasping the fundamental concepts of probability theory, especially in scenarios involving independent events like coin tosses.