Jacob flips a coin and then rolls a six-sided die. He draws a tree diagram to represent this. Find the probability that the die showed a number less than three, given that the coin showed tails.
This question is an example of conditional probability. We are asked to calculate the probability that the die showed a number less than three given that the coins showed tails. It is important to note that the two events of flipping a coin and rolling a die are independent. No matter whether the coin lands on heads or tails, it doesn’t affect the roll of the die.
For any two independent events 𝐴 and 𝐵, the probability of 𝐴 given 𝐵 is simply equal to the probability of 𝐴. This means that, in our question, we simply need to calculate the probability of rolling less than a three on a die. We know that a regular die has the numbers from one to six on each of its faces. The numbers one and two are less than six. This means that the probability of rolling less than three is two out of six or two-sixths. Dividing the numerator and denominator of this fraction by two gives us one-third. The probability that the die showed a number less than three given that the coin showed tails is one-third.
We could also have found this answer directly from the tree diagram. We are told that the coin show tails, so we follow the bottom path. We then know that the die must show either a one or a two. As both of these are equal to one-sixth, we can find the sum of them to calculate our answer. This is equal to two-sixths, which once again simplifies to one-third, confirming our previous answer.