An experiment has four possible outcomes, labeled A to D. The probability of A is P_A = 40% and of B is P_B = 30%. Outcome C is twice as probable as outcome D. What are the probabilities P_C and P_D?

Verified step by step guidance

1. Step 1: Start by understanding the problem. You are given probabilities for outcomes A and B, and a relationship between outcomes C and D. The total probability for all outcomes must sum to 100% (or 1 in decimal form).
2. Step 2: Write the equation for the total probability: Pₐ + PB + PC + PD = 1. Substitute the given values for Pₐ and PB: 0.40 + 0.30 + PC + PD = 1.
3. Step 3: Use the relationship between outcomes C and D. The problem states that outcome C is twice as probable as outcome D, so PC = 2 × PD.
4. Step 4: Substitute PC = 2 × PD into the total probability equation: 0.40 + 0.30 + 2 × PD + PD = 1.
5. Step 5: Combine like terms and solve for PD. Once PD is found, use the relationship PC = 2 × PD to calculate PC.

Key Concepts

Probability Basics

Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1, or as a percentage. The sum of the probabilities of all possible outcomes in a given experiment must equal 1 (or 100%). Understanding this principle is crucial for solving problems involving multiple outcomes.

Setting Up Equations

In problems involving probabilities, it is often necessary to set up equations based on the relationships between different outcomes. For instance, if one outcome is defined in terms of another (like C being twice as probable as D), this relationship can be expressed mathematically to find unknown probabilities.

Solving for Unknowns

Once the equations are established, solving for unknown probabilities involves algebraic manipulation. This may include substituting known values and simplifying equations to isolate the variable of interest. Mastery of these techniques is essential for determining the probabilities of outcomes in complex scenarios.