Understanding Probability with Coin Tosses
Probability is the measure of the likelihood that an event will occur. When dealing with independent events like tossing coins, we can use the concept of a Sample Space to determine all possible outcomes.
The Problem
We are asked to find the probability of getting all heads when three coins are tossed simultaneously.
Step-by-Step Solution
1. Identify the Sample Space ($S$)
Each coin has 2 possible outcomes: Head ($H$) or Tail ($T$). Since there are 3 coins, the total number of outcomes is $2 \times 2 \times 2 = 2^3 = 8$.
The sample space $S$ consists of: $S = \{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}$
2. Identify the Favorable Outcomes ($E$)
We are looking for the outcome where all coins show heads. Looking at our sample space, there is only one such outcome: $E = \{HHH\}$ So, the number of favorable outcomes is $n(E) = 1$.
3. Calculate the Probability ($P$)
The formula for probability is: $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$
Substituting our values: $P(E) = \frac{1}{8}$
Intuition
Since each coin is independent, the probability of getting a head on one coin is $\frac{1}{2}$. The probability of getting heads on three independent coins is the product of their individual probabilities: $P(H, H, H) = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}$.