Math Problem Statement

The following table shows the frequency of outcomes when two distinguishable coins were tossed 4,800 times and the uppermost faces were observed. HINT [See Example 2.] Outcome HH HT TH TT Frequency 1,300 1,150 1,400 950 What is the relative frequency that heads comes up at least once? (Round your answer to four decimal places.)

Solution

To find the relative frequency that heads comes up at least once, let's first identify the relevant outcomes and frequencies.

The possible outcomes when two coins are tossed are:

  • HH (both heads)
  • HT (heads and tails)
  • TH (tails and heads)
  • TT (both tails)

We are interested in the outcomes where at least one head comes up. These outcomes are:

  • HH (both heads)
  • HT (heads and tails)
  • TH (tails and heads)

Now, let's calculate the total frequency for these outcomes:

Total frequency of heads at least once=Frequency of HH+Frequency of HT+Frequency of TH\text{Total frequency of heads at least once} = \text{Frequency of HH} + \text{Frequency of HT} + \text{Frequency of TH} =1,300+1,150+1,400=3,850= 1,300 + 1,150 + 1,400 = 3,850

Next, the total number of trials is 4,800.

The relative frequency that heads comes up at least once is:

Relative frequency=Total frequency of heads at least onceTotal number of trials=3,8504,800\text{Relative frequency} = \frac{\text{Total frequency of heads at least once}}{\text{Total number of trials}} = \frac{3,850}{4,800}

Now, let's compute that:

3,8504,8000.8021\frac{3,850}{4,800} \approx 0.8021

Thus, the relative frequency that heads comes up at least once is approximately 0.8021.

Would you like more details or have any questions? Here are 5 related questions to think about:

  1. What is the probability of getting two tails in this experiment?
  2. How would the relative frequency change if the total trials were different?
  3. What are the odds of getting exactly one head?
  4. Can you use this data to estimate the probability of heads or tails in a single coin toss?
  5. How would you calculate the relative frequency of tails showing up at least once?

Tip: When calculating relative frequencies, always ensure the total counts and event counts are accurate before dividing.

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Math Problem Analysis

Mathematical Concepts

Probability
Relative Frequency

Formulas

Relative frequency = (Number of favorable outcomes) / (Total number of trials)

Theorems

Basic Probability Theorem

Suitable Grade Level

Grades 6-8