Problem 45
Question
A single die is rolled twice. Find the probability of getting: a 2 the first time and a 3 the second time.
Step-by-Step Solution
Verified Answer
The probability of rolling a 2 first and then a 3 is 1/36.
1Step 1: Understand the Basics of Probability
Probability is calculated by dividing the number of desired outcomes by the number of possible outcomes. For a single die, there are 6 possible outcomes on each roll (the numbers 1 through 6).
2Step 2: Calculate the Probability for Each Event
The probability of getting a 2 on the first roll is 1 in 6, and the probability of getting a 3 on the second roll is also 1 in 6. So, \(P(2)\) = 1/6 and \(P(3)\) = 1/6.
3Step 3: Calculate the Combined Probability
Since these are independent events (the outcome of the first roll doesn't affect the second), the combined probability is calculated by multiplying the probabilities of the individual events. So, \(P(2 and 3)\) = \(P(2)\) * \(P(3)\) = (1/6) * (1/6) = 1/36.
Key Concepts
Independent eventsProbability of compound eventsRolling a die
Independent events
When discussing probability, it's important to understand what independent events are. Independent events are those where the outcome of one event does not affect the outcome of another. In simpler terms, knowing the result of one does not give you any new information about the result of the other.
For example, when you roll a die twice, what happens in the first roll does not influence what happens in the second roll. Each roll is a fresh start with no memory of what happened before. This independence is a crucial aspect when calculating probabilities for multiple events, as it allows us to multiply their probabilities to find the probability of both events occurring.
In our exercise, getting a 2 on the first roll and a 3 on the second roll are two independent events. This means the probability of getting a 2 is the same no matter what you rolled before.
For example, when you roll a die twice, what happens in the first roll does not influence what happens in the second roll. Each roll is a fresh start with no memory of what happened before. This independence is a crucial aspect when calculating probabilities for multiple events, as it allows us to multiply their probabilities to find the probability of both events occurring.
In our exercise, getting a 2 on the first roll and a 3 on the second roll are two independent events. This means the probability of getting a 2 is the same no matter what you rolled before.
Probability of compound events
The probability of compound events refers to the chance of two or more events happening together. For independent events, we calculate their combined probability by multiplying the probability of each individual event. This is possible because of their independence, making it simple to find the likelihood of both occurring.
Given our exercise, the probability of rolling a 2 first and then a 3 is a compound event. We calculated the probability of rolling a 2 as \( \frac{1}{6} \) and similarly, for a 3 it is \( \frac{1}{6} \). These probabilities are multiplied to find the combined probability pf getting a 2 followed by a 3, which is \( \frac{1}{36} \).
This approach works whenever dealing with independent events. Just remember to ensure the events don’t affect each other for this rule to apply. Otherwise, adjustments may be needed if events are not independent.
Given our exercise, the probability of rolling a 2 first and then a 3 is a compound event. We calculated the probability of rolling a 2 as \( \frac{1}{6} \) and similarly, for a 3 it is \( \frac{1}{6} \). These probabilities are multiplied to find the combined probability pf getting a 2 followed by a 3, which is \( \frac{1}{36} \).
This approach works whenever dealing with independent events. Just remember to ensure the events don’t affect each other for this rule to apply. Otherwise, adjustments may be needed if events are not independent.
Rolling a die
Rolling a die is a common scenario to explain basic probability concepts. A typical die has six faces, numbered from 1 to 6. Each face has an equal chance of landing up when you roll it, assuming the die is fair.
Understanding this basic setup helps when dealing with more complex probability questions involving dice. This includes situations like rolling a pair of dice or rolling a die multiple times, as seen in our exercise. For each roll, the probability calculations rest on the assumption that each face is equally likely, making dice an ideal tool for probability practice.
- There are 6 possible outcomes on each roll.
- The chance of any specific number is \( \frac{1}{6} \).
Understanding this basic setup helps when dealing with more complex probability questions involving dice. This includes situations like rolling a pair of dice or rolling a die multiple times, as seen in our exercise. For each roll, the probability calculations rest on the assumption that each face is equally likely, making dice an ideal tool for probability practice.
Other exercises in this chapter
Problem 44
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