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Calculate the probability of compound events

Compound-event probability is determined by analyzing an organized sample space and combining outcomes: probabilities are added for mutually exclusive alternatives and multiplied for independent successive events, while dependent outcomes require accounting for how earlier outcomes change later ones. Tables, lists, and tree diagrams represent these relationships and support answers as fractions, decimals, or percents; advanced conditional-probability formulas, permutations, and combinations are not included.

Detailed Explanation: Calculate the probability of compound events

A compound event combines two or more events. Use these rules:

  • Multiply probabilities for events that happen in sequence.
  • Add probabilities for different, mutually exclusive ways the event can happen.
  • For dependent events, update the total after the first outcome.

Example: A bag contains 3 red marbles and 2 blue marbles. You draw 2 marbles without replacing the first one. What is the probability of drawing exactly 1 red and 1 blue?

There are two possible orders:

  1. Red, then blue: (RB)(RB)
  2. Blue, then red: (BR)(BR)

These outcomes cannot happen at the same time, so we will add their probabilities.

Step 1: Find the probability of (RB)(RB).

The probability of red first is 35\frac{3}{5}. After taking a red marble, 4 marbles remain, including 2 blue marbles. So the probability of blue second is 24\frac{2}{4}.

P(RB)=35×24=620=310P(RB)=\frac{3}{5}\times\frac{2}{4}=\frac{6}{20}=\frac{3}{10}

Step 2: Find the probability of (BR)(BR).

The probability of blue first is 25\frac{2}{5}. After taking a blue marble, 4 marbles remain, including 3 red marbles. So the probability of red second is 34\frac{3}{4}.

P(BR)=25×34=620=310P(BR)=\frac{2}{5}\times\frac{3}{4}=\frac{6}{20}=\frac{3}{10}

Step 3: Add the two possible orders.

P(exactly 1 red and 1 blue)=310+310=610=35P(\text{exactly 1 red and 1 blue}) =\frac{3}{10}+\frac{3}{10} =\frac{6}{10} =\frac{3}{5}

So, the probability is

35=60%\boxed{\frac{3}{5}=60\%}

The probabilities were multiplied within each order because both draws must happen. Then the two orders were added because either order gives exactly 1 red and 1 blue.

Learn by doing: Calculate the probability of compound events

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