Question

Tariq and Lucia are finalists in a baking competition. For the final round, each of them will randomly select a card without replacement that will reveal what their primary flavor μst be. Here are the available cards:

a) Chocolate

b) Vanilla

c) Strawberry

d) Lemon

Choose a card, and explain how the selected flavors might influence their final recipes.

Answer 1

The problem involves calculating dependent probabilities of selecting different flavored cookies from a box using a tree diagram. The selections are dependent because removing one cookie affects the chance of picking a particular flavor next. Two methods, including the complement rule and analyzing tree branches, can determine the probability of picking different flavors.

Step-by-step explanation:

Understanding Probabilities in Cookie Selection

The problem you're asking about involves a box with three chocolate and seven butter cookies and the probabilities of picking certain flavors when selecting two cookies without replacement. Let me explain how to approach this question step by step.

Firstly, to address part a), you would create a probability tree diagram to represent all possible outcomes for the two selections. To construct the tree, start with two branches for the first cookie selection: one for chocolate (3 chances out of 10) and one for butter (7 chances out of 10). Then for each of those branches, draw additional branches for the second selection, keeping in mind that one cookie has already been removed from the box. The probabilities on these branches will need to be adjusted accordingly.

In part b), the probabilities for the flavor of the second cookie are not independent of the first selection, because the choice of the first cookie affects the remaining number of cookies in the box and thus changes the probabilities for the second selection.

For part e), to find the probability P(T) that the cookies selected were different flavors, we can use two methods: the complement rule or the tree branches. Using the complement rule involves taking one minus the probability that both selections are the same. The tree diagram shows individual probabilities for each scenario of picking two cookies of different flavors. Both methods should yield the same result.

Lastly, part f) asks us to calculate the probability P(U) that the second cookie selected is a butter cookie. This can be found by considering the adjusted probabilities for each possible first pick in the tree diagram and multiplying them by the probabilities of picking a butter cookie subsequently from the remaining cookies.

Overall, the two key concepts in this question are dependent probabilities and the use of tree diagrams to organize outcomes.