Math Problem Statement
Solution
The problem involves a set of trials where both a number cube (with faces numbered 1 to 6) and a coin (with heads and tails) are rolled/flipped. The data from 60 trials is summarized in the table provided. You are asked to calculate experimental and theoretical probabilities based on this data. Let's break down the steps to solve the problem.
Part (a): Experimental Probability
You need to find the experimental probability of the event: "both rolling an odd number and flipping tails in a single trial."
- Outcome that involves an odd number and tails (T):
- Odd numbers on the die: 1, 3, 5.
- Tails (T) on the coin.
From the table, the relevant outcomes are:
- 1T (3 trials)
- 3T (4 trials)
- 5T (6 trials)
The total number of trials is 60. The total number of trials where the outcome is "odd number and tails" is the sum of the trials for 1T, 3T, and 5T:
The experimental probability is given by:
So, the experimental probability is approximately 0.217.
Part (b): Theoretical Probability
Assuming both the die and coin are fair, you can calculate the theoretical probability.
-
The probability of rolling an odd number on a fair 6-sided die is:
-
The probability of flipping tails on a fair coin is:
The theoretical probability of both rolling an odd number and flipping tails is the product of the two individual probabilities:
Part (c): True Statement about Experimental Probability
The question asks you to choose the correct statement about experimental probability. Here’s the reasoning for each option:
- With a small number of trials, it's not surprising when the experimental probability is much more or much less than the theoretical probability.
This is because small sample sizes can lead to fluctuations and deviations from the theoretical probability. Larger sample sizes tend to give probabilities that are closer to the theoretical values.
Let me know if you need further clarification or help with additional parts!
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Math Problem Analysis
Mathematical Concepts
Probability
Experimental Probability
Theoretical Probability
Formulas
P(odd number and tails) = Number of favorable outcomes / Total trials
P(odd number) = 3/6
P(tails) = 1/2
P(odd number and tails) = P(odd number) * P(tails)
Theorems
Probability of independent events
Suitable Grade Level
Grades 6-8
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