in a game of lotto six balls from a set of 40 balls, numbered from 1 to 40 are selected. find the probability that the number is drawn in exactly two of the next five games

To solve this problem, we need to determine the probability of drawing a number in exactly **two** of the next five games in a lotto where six balls are selected from a set of 40.

We will break this down step by step:

### Step 1: Probability of a number being drawn in one game
In one game, six balls are drawn from a total of 40. The probability that a particular number (say \(x\)) is drawn can be calculated as:

\[
P(\text{number is drawn in one game}) = \frac{6}{40} = 0.15
\]

Similarly, the probability that the number is **not** drawn in one game is:

\[
P(\text{number is not drawn in one game}) = 1 - 0.15 = 0.85
\]

### Step 2: Probability of drawing the number in exactly two out of five games
This is a binomial probability problem, where:
- We have 5 independent games (trials).
- The probability of success (the number being drawn in a game) is \(p = 0.15\).
- The probability of failure (the number not being drawn) is \(1 - p = 0.85\).
- We want exactly **two successes** in the five games.

The binomial probability formula is:

\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k}
\]

Where:
- \(n = 5\) (the number of games),
- \(k = 2\) (the number of games we want the number to be drawn),
- \(p = 0.15\) (the probability of drawing the number in one game).

Plugging in the values:

\[
P(X = 2) = \binom{5}{2} (0.15)^2 (0.85)^{5-2}
\]

First, calculate the binomial coefficient \(\binom{5}{2}\):

\[
\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10
\]

Now calculate the full expression:

\[
P(X = 2) = 10 \times (0.15)^2 \times (0.85)^3
\]

We will now compute this value.

### Step 3: Calculate the probability
Let's calculate it step by step.

\[
P(X = 2) = 10 \times (0.15)^2 \times (0.85)^3
\]

Let's compute the result.The probability that a number is drawn in exactly two of the next five games is approximately **0.138** or **13.8%**.

Would you like further details or have any questions?

Here are 5 related questions you might find interesting:
1. What is the probability that the number is drawn in exactly one of the next five games?
2. How would the probability change if the number of balls increased to 50 instead of 40?
3. What is the probability that the number is not drawn in any of the next five games?
4. How does the probability change if the number is drawn in three of the next five games?
5. What if we wanted the probability of the number being drawn at least once in the next five games?

**Tip**: Binomial probability distribution is useful when you are dealing with a fixed number of independent trials, each with only two outcomes (success or failure).

In a game of lotto, six balls from a set of 40 balls, numbered from 1 to 40, are selected. Find the probability that the number is drawn in exactly two of the next five games.

Discover how to calculate the probability of drawing a specific number in exactly two of the next five lotto games. This problem involves binomial probability distribution, including key formulas and concepts. Ideal for students in Grades 10-12.

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