Of course! Let's break down the problem step by step to determine the probability of each event.
1. Determining the Probability that Ling Selects a Raspberry Jam Doughnut:
- There are a total of 16 doughnuts.
- Out of these, 9 are raspberry jam doughnuts.
The probability that Ling selects a raspberry jam doughnut is the ratio of the number of raspberry jam doughnuts to the total number of doughnuts.
[tex]\[
\text{Probability (Raspberry Jam for Ling)} = \frac{\text{Number of Raspberry Jam Doughnuts}}{\text{Total Number of Doughnuts}} = \frac{9}{16} = 0.5625
\][/tex]
2. Determining the Probability that Alexander Selects a Lemon Curd Doughnut Given Ling has Chosen a Raspberry Jam Doughnut:
- After Ling selects a raspberry jam doughnut, one doughnut is removed from the tray. So, the total number of doughnuts now is 15.
- The number of lemon curd doughnuts remains the same, which is 7.
The probability that Alexander selects a lemon curd doughnut from the remaining 15 doughnuts is the ratio of the number of lemon curd doughnuts to the remaining number of doughnuts.
[tex]\[
\text{Probability (Lemon Curd for Alexander)} = \frac{\text{Number of Lemon Curd Doughnuts}}{\text{Remaining Number of Doughnuts}} = \frac{7}{15} = 0.4667
\][/tex]
3. Combined Probability of Both Events Happening:
- These two events are independent, so the probability of both events occurring is the product of their individual probabilities.
[tex]\[
\text{Combined Probability} = \text{Probability (Raspberry Jam for Ling)} \times \text{Probability (Lemon Curd for Alexander)} = 0.5625 \times 0.4667 = 0.2625
\][/tex]
So, the detailed step-by-step solution provides us with:
- The probability that Ling selects a raspberry jam doughnut: 0.5625
- The probability that Alexander selects a lemon curd doughnut given that Ling has already selected a raspberry jam doughnut: 0.4667
- The combined probability of both events happening: 0.2625
