Skill: Calculate probabilities using counting techniques

Probability using counting techniques follows this structure:

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

Use:

• Combinations when order does not matter: $\binom{n}{r}$
• 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 $5$ red balls and $4$ 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 $3$ balls from $9$ total balls. Since the order of selection does not matter, use a combination:

$$\text{Total outcomes}=\binom{9}{3}=84.$$

Step 2: Break the favorable outcomes into cases

“At least two red” means either:

• exactly $2$ red and $1$ blue, or
• exactly $3$ red.

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

Case 1: Exactly $2$ red and $1$ blue

Choose $2$ of the $5$ red balls and $1$ of the $4$ blue balls:

$$\binom{5}{2}\binom{4}{1}=10\cdot4=40.$$

The multiplication is used because both choices must occur.

Case 2: Exactly $3$ red

Choose $3$ of the $5$ red balls:

$$\binom{5}{3}=10.$$

Therefore,

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

Step 3: Find the probability

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

So, the probability is

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

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.