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Difficulty: MediumExperimental and Theoretical Probability

A container holds a large number of red, blue, and yellow counters. The theoretical probability of selecting a red counter at random is 25\frac{2}{5}, and the theoretical probability of selecting a blue counter is 13\frac{1}{3}. In an experiment where a counter is drawn and replaced 300300 times, a yellow counter is selected 7272 times. What is the absolute difference between the theoretical expected number of yellow counters and the experimental frequency of yellow counters obtained?

  1. 88Answer
  2. B
    148148
  3. C
    2828
  4. D
    8080

Answer

The absolute difference between the theoretical expected number of yellow counters and the experimental frequency is 88.
The theoretical probability of drawing a yellow counter is 1−(25+13)=4151 - (\frac{2}{5} + \frac{1}{3}) = \frac{4}{15}. For 300300 trials, the theoretical expected count of yellow counters is 415×300=80\frac{4}{15} \times 300 = 80. Subtracting the experimental observed count of 7272 from 8080 gives an absolute difference of 88.

Step-by-Step Solution

1
Calculate the theoretical probability of drawing a yellow counter.
P(Yellow)=1−(P(Red)+P(Blue))=1−(25+13)=1−1115=415P(\text{Yellow}) = 1 - \left(P(\text{Red}) + P(\text{Blue})\right) = 1 - \left(\frac{2}{5} + \frac{1}{3}\right) = 1 - \frac{11}{15} = \frac{4}{15}
The sum of probabilities of all mutually exclusive outcomes in the sample space must equal 11.
2
Calculate the theoretical expected frequency of yellow counters in 300300 trials.
Expected frequency=P(Yellow)×Total trials=415×300=80\text{Expected frequency} = P(\text{Yellow}) \times \text{Total trials} = \frac{4}{15} \times 300 = 80
The expected number of occurrences of an outcome is the product of its theoretical probability and the number of trials.
3
Find the absolute difference between expected frequency and experimental frequency.
|80 - 72| = 8
The question asks for the difference between the theoretical expected value (8080) and the actual experimental result (7272).

Key Concept

Experimental and Theoretical Probability
Estimated Time:1m 30s
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Experimental and Theoretical Probability

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