Math Problem Statement

There is a number cube with faces numbered 1 to 6. There is also a coin with one side marked as heads and the other tails. As a trial of an experiment, the number cube was rolled and the coin flipped. The number (1 to 6) from the roll and the side (H for heads and T for tails) of the coin from the flip were recorded. Here is a summary of the data from 60 trials. Use the data to find the experimental probability of this event: both rolling an odd number and flipping tails in a single trial. Round your answer to the nearest thousandth.

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."

  1. 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: 3+4+6=13 trials3 + 4 + 6 = 13 \text{ trials}

The experimental probability is given by: P(odd number and tails)=Number of favorable outcomesTotal trials=13600.217P(\text{odd number and tails}) = \frac{\text{Number of favorable outcomes}}{\text{Total trials}} = \frac{13}{60} \approx 0.217

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.

  1. The probability of rolling an odd number on a fair 6-sided die is: P(odd number)=36=12P(\text{odd number}) = \frac{3}{6} = \frac{1}{2}

  2. The probability of flipping tails on a fair coin is: P(tails)=12P(\text{tails}) = \frac{1}{2}

The theoretical probability of both rolling an odd number and flipping tails is the product of the two individual probabilities: P(odd number and tails)=12×12=14=0.250P(\text{odd number and tails}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} = 0.250

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