A student asks, “What’s wrong with the argument that the probability of rolling a double 6 in two rolls of a die is 1/3 because 1/6 + 1/6 = 1/3?" Write an explanation of your understanding of the student’s misconception.
The student's argument that the probability of rolling a double 6 in two rolls of a die is 1/3 because 1/6 + 1/6 equals 1/3 is based on a misconception about probability.
To understand why this is incorrect, let's break down the concept of probability. In this case, we are rolling a fair six-sided die, where each face has an equal chance of coming up.
When rolling a single die, the probability of rolling a 6 is indeed 1 out of 6, or 1/6. Since there are two rolls involved, the student's intuition might lead them to think that they can simply add the probabilities together.
However, the probability of rolling a double 6 means getting two sixes in a row on consecutive rolls. To determine the probability of two independent events happening together, we multiply their individual probabilities. In this case, the probability of rolling a 6 in the first roll is 1/6, and the probability of rolling another 6 in the second roll is also 1/6.
To calculate the probability of both events occurring, we multiply these probabilities: (1/6) * (1/6) = 1/36. Therefore, the probability of rolling a double 6 is actually 1/36, not 1/3.
The misconception in the student's argument arises from incorrectly assuming that probabilities can be added together in this scenario. However, in the case of independent events like rolling a die twice, probabilities are multiplied, not added.
It's important to keep in mind that probability is a mathematical concept, and understanding how to properly calculate it is crucial to avoid such misconceptions.