Understanding Basic Probability with Dice
Probability is the measure of the likelihood that an event will occur. It is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes in a sample space.
The Problem Setup
When you throw two standard six-sided dice, each die has 6 faces numbered 1 through 6.
- Total Outcomes: Since each die is independent, the total number of possible outcomes is $6 \times 6 = 36$.
Step-by-Step Calculation
1. Identify Odd Digits on a Die
On a single die, the possible numbers are {1, 2, 3, 4, 5, 6}. The odd digits are {1, 3, 5}. There are 3 odd digits out of 6 total possibilities for one die.
2. Probability for One Die
The probability of getting an odd digit on one die is: $P(Odd) = \frac{3}{6} = \frac{1}{2}$
3. Probability for Two Dice
Since the dice are independent, we can multiply their individual probabilities: $P(Both Odd) = P(Odd \text{ on Die 1}) \times P(Odd \text{ on Die 2})$ $P(Both Odd) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$
Alternatively, we can list the favorable pairs: {(1,1), (1,3), (1,5), (3,1), (3,3), (3,5), (5,1), (5,3), (5,5)} There are 9 favorable outcomes out of 36 total outcomes: $P = \frac{9}{36} = \frac{1}{4}$
Final Answer
The probability of getting both odd digits is $\frac{1}{4}$ or $0.25$ (25%).