Theoretical Probability Statistics Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

medium
A bag has 4 red and 6 blue balls. One ball is drawn, replaced, and a second is drawn. What is the probability both are red?

Solution

  1. 1
    Step 1: P(1st red)=410=25P(\text{1st red}) = \frac{4}{10} = \frac{2}{5}. With replacement, P(2nd red)=25P(\text{2nd red}) = \frac{2}{5}.
  2. 2
    Step 2: Since draws are independent: P(both red)=25×25=425P(\text{both red}) = \frac{2}{5} \times \frac{2}{5} = \frac{4}{25}.

Answer

425\frac{4}{25}
With replacement, each draw is independent, so we multiply the individual probabilities. Without replacement, the probabilities would change for the second draw.

About Theoretical Probability

Theoretical probability is the expected probability of an event calculated by mathematical reasoning about equally likely outcomes, without conducting experiments. It equals the number of favorable outcomes divided by the total number of possible outcomes.

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More Theoretical Probability Examples

Example 1 medium

Two fair coins are tossed. Find the theoretical probability of getting exactly one head.

Example 2 medium

A fair die and a fair coin are used together. What is the probability of rolling a 3 AND getting hea

Example 4 medium

Two fair six-sided dice are rolled. What is the theoretical probability that the sum is 7?

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