Question

Three coins are tossed; what is the probability that two are heads and one tails? That the first two are heads and the third tails? If at least two are heads, what is the probability that all are heads?

Short answer

Probabilities are \( \frac{3}{8} \), \( \frac{1}{8} \), and \( \frac{1}{4} \).

Step-by-step solution

Determine Total Possible Outcomes

When three coins are tossed, each coin can result in two possible outcomes: heads or tails. Therefore, the total number of possible outcomes is calculated as: \[ 2^3 = 8 \] So, there are 8 possible outcomes.

List All Possible Outcomes

Enumerate all the possible results of three coin tosses:- HHH- HHT- HTH- HTT- THH- THT- TTH- TTT

Count Outcomes with Two Heads and One Tail

From the list of outcomes, identify the ones with exactly two heads and one tail:- HHT- HTH- THHThere are 3 outcomes that meet this criterion.

Calculate Probability of Two Heads and One Tail

The probability of two heads and one tail is the ratio of favorable outcomes to the total outcomes:\[ \frac{3}{8} \]

Count Outcomes with First Two Heads and Third Tail

Identify the outcomes where the first two coins are heads and the third is a tail:- HHTThere is 1 outcome that meets this criterion.

Calculate Probability of First Two Heads and Third Tail

The probability of the first two coins being heads and the third a tail is the ratio of favorable outcomes to the total outcomes:\[ \frac{1}{8} \]

Count Outcomes with At Least Two Heads

Identify the outcomes with at least two heads:- HHH- HHT- HTH- THHThere are 4 outcomes that meet this criterion.

Calculate Probability that All Are Heads Given At Least Two Heads

Among the outcomes with at least two heads (HHH, HHT, HTH, THH), the one where all are heads is HHH. The probability is given by the ratio of the number of outcomes where all are heads (1) to the number of outcomes with at least two heads (4):\[ \frac{1}{4} \]

Key concepts

Probability Theory

In probability theory, we study the likelihood of certain events occurring. An event is simply an outcome or a set of outcomes of an experiment, such as tossing a coin. When we toss three coins, each coin has two potential outcomes: heads (H) or tails (T). Therefore, there are a total of We can list these outcomes as:

• HHH
• HHT
• HTH
• HTT
• THH
• THT
• TTH
• TTT

.

.Because there are 8 possible outcomes in total. When calculating probabilities, we use the formula where favorable outcomes are the outcomes that satisfy the event condition and total possible outcomes are all the possible outcomes of the experiment.

Combinatorics

Combinatorics involves counting, arranging, and combining objects. In this problem, we count specific outcomes of coin tosses, such as having exactly two heads and one tail. To solve this, we list all outcomes:

• HHH
• HHT
• HTH
• HTT
• THH
• THT
• TTH
• TTT

Next, we identify which outcomes have two heads and one tail. These outcomes are:

• HHT
• HTH
• THH

There are 3 such outcomes. To calculate the probability, we use the ratio of these favorable outcomes to the total number of outcomes. In this case, it is Using combinatorics makes counting manageable and ensures accuracy.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It helps us refine our probability calculations. For instance, we might want to know the probability that all three coins show heads given that at least two do. First, we find the favorable outcomes where at least two are heads:

• HHH
• HHT
• HTH
• THH

There are 4 such outcomes. Out of these, only one outcome has all three heads (HHH). So, the conditional probability is . We only consider the subset of outcomes where the condition (at least two heads) is satisfied, which refines our count for more accurate calculation.

Discrete Mathematics

Discrete mathematics, which deals with countable structures, applies to scenarios like our coin-tossing problem. Here, we work with discrete outcomes (heads or tails). We list and categorize these outcomes to solve probability problems. The total number of possible outcomes when three coins are tossed is a power of 2 (since each coin has 2 states), specifically Listing out all the discrete outcomes allows us to methodically count and compute the probabilities. For example, if we seek the probability of getting heads in the first two tosses and tails in the third, we identify the single outcome (HHT) from our list. There are 8 outcomes in total, making the probability . Discrete mathematics provides essential tools for breaking down and solving such problems step-by-step, ensuring clarity and accuracy.