In which scenario is the experimental probability equal to the theoretical probability?
A.) A coin is flipped six times and the head appears each time.
B.)A coin is flipped six times and the head appears four times.
C.)A coin is flipped six times and the head appears five times.
D.)A coin is flipped six times and the head appears three times.
D.) A coin is flipped six times and the head appears three times.
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Theoretical probability is what we expect to happen, or what should happen in theory. It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
Experimental probability is what actually happens when we perform an experiment or an observation. It is calculated by dividing the number of favorable outcomes by the total number of trials.
In this scenario, the theoretical probability of getting heads on a coin flip is 1/2, or 50%. The probability of getting exactly three heads in six flips can be calculated using the binomial probability formula P(x=3) = (6 choose 3) * (1/2)^3 * (1/2)^3 = 20/64 = 0.3125, or approximately 31.25%.
If we flip the coin six times and record the results, the experimental probability of getting three heads should be close to the theoretical probability of 31.25%. Therefore, the experimental probability is equal to the theoretical probability in this scenario.
To determine when the experimental probability is equal to the theoretical probability, we first need to understand what these terms mean.
Experimental probability is the probability of an event occurring based on actual outcomes from an experiment or trial. It is calculated by dividing the number of times the event occurs by the total number of trials.
Theoretical probability, on the other hand, is the probability of an event occurring based on logical reasoning or mathematical calculations. It is determined by dividing the number of favorable outcomes by the total number of possible outcomes.
In this scenario, we are flipping a coin six times and interested in the appearance of heads. The theoretical probability of getting a head on one coin flip is 1/2 since there are two equally likely outcomes (head or tail). Thus, the theoretical probability of getting a head on six flips would be (1/2)^6 or 1/64.
Now let's look at the options:
A.) A coin is flipped six times, and the head appears each time.
In this case, if the head appears each time, the experimental probability would be 1 since all trials resulted in heads. However, the theoretical probability is only 1/64. Therefore, the experimental probability in scenario A does not equal the theoretical probability.
B.)A coin is flipped six times, and the head appears four times.
In this case, the experimental probability can be calculated by dividing the number of heads by the total number of trials. Since we have four heads out of six trials, the experimental probability would be 4/6, which simplifies to 2/3. The theoretical probability is still 1/64. Hence, the experimental probability in scenario B does not equal the theoretical probability.
C.)A coin is flipped six times, and the head appears five times.
Similar to scenario B, the experimental probability would be calculated as 5/6 since there are five heads out of six trials. Again, the theoretical probability remains 1/64. Thus, the experimental probability in scenario C does not equal the theoretical probability.
D.)A coin is flipped six times, and the head appears three times.
In this case, the experimental probability would be 3/6 or 1/2 since there are three heads out of six trials. The theoretical probability, as previously mentioned, is 1/64. Surprisingly, the experimental probability in this scenario does equal the theoretical probability. Therefore, the correct answer is option D.
In summary:
- The experimental probability is the same as the theoretical probability when a coin is flipped six times, and the head appears three times (option D).