Problem 35 - Algebra 2

Question

Determine whether the events are independent or dependent. Then find the probability. When Ramon plays basketball, he makes an average of two out of every three foul shots he takes. What is the probability that he will make the next three foul shots in a row?

Step-by-Step Solution

Answer

The events are independent, and the probability is \( \frac{8}{27} \).

Step 1: Define Independent or Dependent Events

Independent events are those whose outcomes do not affect each other, while dependent events have outcomes that are influenced by previous events. Since making one basketball shot does not affect the next, these are independent events.

Step 2: Determine Probability of Single Event

The probability that Ramon makes one foul shot is given as two out of every three shots. Therefore, the probability is \( P( ext{make}) = \frac{2}{3} \).

Step 3: Calculate Probability of Consecutive Events

Since the events are independent, the probability of making three shots in a row is the product of the probabilities of making each single shot. This is calculated as follows: \[ P(3 ext{ shots}) = P( ext{make}) \times P( ext{make}) \times P( ext{make}) = \left(\frac{2}{3}\right)^3. \]

Step 4: Perform the Calculation

Execute the multiplication from Step 3: \[ \left(\frac{2}{3}\right)^3 = \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} = \frac{8}{27}. \] Thus, the probability that Ramon will make the next three shots is \( \frac{8}{27} \).

Key Concepts

Probability

Probability helps us understand how likely an event is to occur. It's like assigning a number to express the chance of something happening. This number is always between 0 and 1. For example, 0 means the event will definitely not happen, and 1 means it definitely will. If you have a 50% chance, that is expressed as 0.5 in probability terms.
To find the probability of a specific event, you use the formula:

• Probability of an event = (Favorable outcomes) / (Total possible outcomes)

For probability calculations, knowing how to set up the problem by clearly identifying the favorable and total outcomes is key. In our basketball example, every shot also gives us a chance to calculate this by using the figures we've been given.

Multiplication of Probabilities

Multiplication of probabilities comes into play with independent events. When you want to calculate the probability of multiple events happening in succession, you multiply the probabilities of each individual event. This only applies when events do not affect each other's outcomes, making them independent.
Consider making three basketball shots in a row, where each event's outcome does not change the others. If the probability of making one shot is \( \frac{2}{3} \), then to find the probability of making all three, calculate:

• \( \text{Probability of 3 shots} = \left( \frac{2}{3} \right) \times \left( \frac{2}{3} \right) \times \left( \frac{2}{3} \right) = \left( \frac{2}{3} \right)^3 \)
• Once calculated, it equals \( \frac{8}{27} \).

In this example, each foul shot is an independent event, so using multiplication allows us to find the combined probability effectively. This is a key technique in working with probabilities of multiple independent events.

Independent versus Dependent Events

Understanding whether events are independent or dependent is crucial in probability.

• Independent Events: The outcome of one event does not affect or change the outcome of another. For instance, flipping a coin or shooting a basketball repeatedly is independent because previous results don’t alter future results.
• Dependent Events: The outcome of one event influences or changes the outcome of another. Consider drawing a card from a deck and not replacing it; the outcome affects what can occur next.

In our exercise, each basketball shot is independent, meaning the outcome of each shot doesn’t change the probability of the next. Recognizing this helps in choosing the correct method for calculating probabilities. Being aware of the difference ensures you correctly apply either the multiplication rule for independent events or adjust your calculations to account for dependencies when they are present.