Question

PROBLEM SOLVING You play a game that involves drawing three numbers from a hat. There are 25 pieces of paper numbered from 1 to 25 in the hat. Each number is replaced after it is drawn. Find the probability that you will draw the 3 on your first draw and a number greater than 10 on your second draw.

Short answer

The probability of drawing a 3 on the first draw and a number greater than 10 on the second draw is 0.024 or 2.4%.

Step-by-step solution

Find the probability of the first event

The first event is drawing a 3. As there are 25 pieces of paper in the hat, each numbered from 1- 25, the probability of drawing a 3 is given as the ratio of the number of successful outcomes (1, i.e., drawing a 3) to the total number of possible outcomes(25). In mathematical form, it can be written as \( \frac{1}{25} = 0.04 \). That is, there is a 4% chance to draw a 3 with the first draw.

Find the probability of the second event

The second event is drawing a number that is greater than 10 in the second draw. As each number is replaced after being drawn, the total number of possible outcomes remains 25 for the second draw as well. The numbers greater than 10 are from 11 to 25, and there are 15 such numbers. So, the probability of drawing a number greater than 10 is \( \frac{15}{25} = 0.6 \). That is, there is a 60% chance to draw a number greater than 10 with the second draw.

Calculate the joint probability of the combined events

As the two events are independent, the probability that both events will occur is simply the product of the individual probabilities. Thus, the probability of drawing a 3 in the first draw and a number greater than 10 in the second draw is \( 0.04 * 0.6 = 0.024 \). That is, there is a 2.4% chance that both conditions will be met in two successive draws.

Key concepts

Independent Events

In probability, independent events are occurrences where the outcome of one event does not influence the outcome of another. This is important when you're dealing with multiple events and want to determine how likely it is that they all happen.

For instance, imagine drawing a number from a hat, putting it back, and then drawing again. The probability of each draw is unaffected by the previous results. If you are curious whether two events, like drawing a 3 and then a number greater than 10, are independent, ask yourself: Does the result of the first draw change the probability of the second draw? If the answer is no, they are independent.

• Example: Drawing a card, replacing it, and drawing again.
• Not Example: Drawing a card and not replacing it, as it affects subsequent draws.

Joint Probability

Joint probability refers to the likelihood of two events occurring together. To find this, especially when dealing with independent events, you multiply the probability of each event.

Consider determining the probability of drawing a 3 first, and then a number greater than 10, where both occurrences are independent. The process involves multiplying their probabilities:

• Probability of first event: drawing a 3.
• Probability of second event: drawing a number greater than 10.

If these probabilities are 0.04 and 0.6 respectively, the joint probability is: \(0.04 \times 0.6 = 0.024\).

Thus, you find there's a 2.4% chance both conditions are met in sequence.

Replacement in Probability

Replacement is a crucial factor in probability calculations. When a number is drawn and then replaced, the conditions for each draw remain unchanged.

This practice keeps the total number of possible outcomes consistent across all draws, which is essential for maintaining independence between events.

• No Replacement: The pool of numbers changes, affecting probabilities.
• With Replacement: The same pool is used for each draw.

For instance, if you are drawing numbers from a set of 25 and always replace them, each draw has the same 1 in 25 chance for each specific number. If you didn’t replace a number, drawing a specific number on future draws would have different odds. Thus, ensuring replacement allows us to easily compute joint probabilities.

Probability of Multiple Events

When calculating the probability of multiple events happening together, you should consider whether the events are independent or dependent. If independent, you multiply their individual probabilities.

Here's how for independent events:

• Determine each event's probability separately.
• Multiply these probabilities to find the combined chance.

For example, say you want to know the likelihood of drawing a 3, then a number greater than 10. Since each draw is independent due to replacement, you just multiply the probability of a 3 (0.04) with the probability of a number greater than 10 (0.6).

This approach simplifies finding such joint occurrences and is vital for solving complex probability problems efficiently.