Darrin tosses a quarter and a penny 20 times. He gets heads on both coins twice. Compare the theoretical probability of getting heads on both coins with Darrin’s experimental probability of getting heads on both coins. Why might the probabilities differ?
The 4 possible outcomes are HH, HT, TH, TT.
So theoretically, HH should occur 1in 4 times.
1/4 = x/20
x = 5
So why is it actually 2 in 20 when theoretically it should be 5 in 20? One reason is that theoretical probabilities gain more accuracy with larger numbers of results (e.g tossing the coins 100 times). You can likely think of some more reasons.
To compare the theoretical probability of getting heads on both coins with the experimental probability, we'll calculate both.
Theoretical Probability:
The probability of getting heads on a fair coin is 1/2. Since Darrin has two coins, the probability of getting heads on both coins is (1/2) * (1/2) = 1/4.
Experimental Probability:
In the experiment, Darrin tossed the coins 20 times and got heads on both coins twice. To calculate the experimental probability, we divide the number of times heads appeared on both coins by the total number of tosses: 2/20 = 1/10.
Comparing the two probabilities:
Theoretical Probability: 1/4
Experimental Probability: 1/10
The probabilities might differ due to random variation. In theory, when the coins are fair, the probability of getting heads on both coins should be 1/4. However, in practice, Darrin's results may vary due to factors such as the way he tosses the coins, the surface they land on, or slight variations in their weight distribution. These factors can cause the coins to behave differently and lead to deviations in the experimental probability. The more times the experiment is repeated, the more likely the experimental probability will converge to the theoretical probability.
To compare the theoretical and experimental probabilities of getting heads on both coins, we need to understand what each probability represents.
Theoretical probability is based on the assumption that all possible outcomes are equally likely. In this case, we have two coins, so there are four possible outcomes: heads-heads, heads-tails, tails-heads, and tails-tails. Since getting heads on a fair coin has a probability of 1/2, the theoretical probability of getting heads on both coins is (1/2) * (1/2) = 1/4.
On the other hand, experimental probability is calculated based on the actual results obtained through repeated trials, which in this case is Darrin tossing the coins 20 times. Darrin got heads on both coins twice out of 20 trials, so the experimental probability of getting heads on both coins is 2/20 = 1/10.
The theoretical and experimental probabilities differ because experimental probability is based on a limited number of trials, while theoretical probability assumes an infinite number of trials. In this case, 20 trials may not be enough to get results that perfectly reflect the theoretical probabilities.
The differences between theoretical and experimental probabilities can be attributed to randomness and the law of large numbers. In a small number of trials, like Darrin's 20 tosses, there can be significant deviations from the expected probabilities. As the number of trials increases, the experimental probability should approach the theoretical probability.