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Calculate probabilities using counting techniques

Probability can be calculated by identifying a finite sample space, counting outcomes with the addition and multiplication principles, and applying permutations or combinations when order does or does not matter. The learner interprets probability as the ratio of favorable outcomes to equally likely total outcomes, accounts for restrictions such as repeated choices or sampling without replacement, and avoids treating outcomes as equally likely when they are not; advanced continuous probability and more abstract combinatorial generalizations are excluded.

Detailed Explanation: Calculate probabilities using counting techniques

Probability using counting techniques follows this structure:

P(event)=number of favorable outcomesnumber of equally likely outcomes.P(\text{event})=\frac{\text{number of favorable outcomes}}{\text{number of equally likely outcomes}}.

Use:

  • Combinations when order does not matter: (nr)\binom{n}{r}
  • The multiplication principle when choices happen in separate groups: multiply the counts.
  • The addition principle when an event can happen in different, non-overlapping cases: add the counts.

Example

A box contains 55 red balls and 44 blue balls. Three balls are chosen without replacement. What is the probability of choosing at least two red balls?

Step 1: Count all possible outcomes

We choose 33 balls from 99 total balls. Since the order of selection does not matter, use a combination:

Total outcomes=(93)=84.\text{Total outcomes}=\binom{9}{3}=84.

Step 2: Break the favorable outcomes into cases

“At least two red” means either:

  • exactly 22 red and 11 blue, or
  • exactly 33 red.

These cases cannot happen at the same time, so we will add their counts.

Case 1: Exactly 22 red and 11 blue

Choose 22 of the 55 red balls and 11 of the 44 blue balls:

(52)(41)=10⋅4=40.\binom{5}{2}\binom{4}{1}=10\cdot4=40.

The multiplication is used because both choices must occur.

Case 2: Exactly 33 red

Choose 33 of the 55 red balls:

(53)=10.\binom{5}{3}=10.

Therefore,

Favorable outcomes=40+10=50.\text{Favorable outcomes}=40+10=50.

Step 3: Find the probability

P(at least two red)=5084=2542.P(\text{at least two red})=\frac{50}{84}=\frac{25}{42}.

So, the probability is

2542.\boxed{\frac{25}{42}}.

The key decisions were recognizing that the selected group has no order, using combinations, multiplying choices within each case, and adding the separate favorable cases.

Learn by doing: Calculate probabilities using counting techniques

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    Probability Sample Space - Sample Space Definition and Favourable Outcomes Count to Probability

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