4. Probability
Complements
2:52 minutesProblem 4.1.1
Textbook Question
California Lottery Let A denote the event of placing a $1 straight bet on the California Daily 4 lottery and winning. There are 10,000 different ways that you can select the four digits (with repetition allowed) in this lottery, and only one of those four-digit numbers will be the winner. What is the value of P(A)? What is the value of P(Abar)?
Verified step by step guidance1
Step 1: Understand the problem. The event A represents winning the California Daily 4 lottery with a $1 straight bet. There are 10,000 possible four-digit combinations (0000 to 9999), and only one of these combinations is the winning number. The probability of event A, P(A), is the ratio of favorable outcomes (1 winning number) to the total possible outcomes (10,000 combinations).
Step 2: Write the formula for P(A). The probability of an event is calculated as P(A) = (Number of favorable outcomes) / (Total number of outcomes). In this case, P(A) = 1 / 10,000.
Step 3: Define the complement of event A, denoted as AÌ„. The complement of A represents the event of not winning the lottery. The probability of the complement, P(AÌ„), is related to P(A) by the formula P(AÌ„) = 1 - P(A).
Step 4: Substitute the value of P(A) into the formula for P(AÌ„). Using the relationship P(AÌ„) = 1 - P(A), substitute P(A) = 1 / 10,000 to calculate P(AÌ„).
Step 5: Conclude that P(A) represents the probability of winning, and P(AÌ„) represents the probability of not winning. These probabilities should sum to 1, as they are complementary events.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Probability
Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, it quantifies the chance of winning the lottery by placing a specific bet. The probability of an event A, denoted as P(A), is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
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Complement of an Event
The complement of an event A, denoted as AÌ… (A bar), represents all outcomes in which event A does not occur. In the lottery scenario, P(AÌ…) would be the probability of not winning the lottery after placing a bet. The relationship between an event and its complement is given by P(A) + P(AÌ…) = 1, which helps in calculating probabilities when one is known.
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Counting Principles
Counting principles, such as the fundamental counting principle, are used to determine the total number of possible outcomes in a scenario. In the California Daily 4 lottery, there are 10,000 possible combinations of four digits (with repetition allowed), which is crucial for calculating the probabilities of winning and losing. Understanding how to count these outcomes accurately is essential for probability calculations.
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