Introduction to Basic Probability
Probability is the branch of mathematics that quantifies the likelihood of an event occurring. At its heart, it is the ratio of the number of favorable outcomes to the total number of equally likely possible outcomes.
The Problem: Tossing Two Coins
When we toss two coins, we are looking at two independent events. Each coin has two possible outcomes: Heads ($H$) or Tails ($T$).
1. Determining the Sample Space
The sample space ($S$) is the set of all possible outcomes of an experiment. For two coins, we can list them systematically:
- Coin 1 shows Heads, Coin 2 shows Heads: $(H, H)$
- Coin 1 shows Heads, Coin 2 shows Tails: $(H, T)$
- Coin 1 shows Tails, Coin 2 shows Heads: $(T, H)$
- Coin 1 shows Tails, Coin 2 shows Tails: $(T, T)$
Therefore, the sample space is: $S = \{ (H, H), (H, T), (T, H), (T, T) \}$. The total number of outcomes is $4$.
2. Calculating the Probability
We want to find the probability that both coins are heads.
- Favorable outcome: $(H, H)$
- Number of favorable outcomes: $1$
The formula for probability $P(E)$ is: $$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$
Applying this: $$P(\text{both heads}) = \frac{1}{4}$$
Converting to decimals or percentages, we get $0.25$ or $25\%$.