To simulate the probability of a snowy Friday, we need to consider two independent probabilities: the probability of it snowing on the first day of winter and the probability that the first day of winter is a Friday.
Step 1: Understand the given probabilities
1. The probability that it will snow on the first day of winter is given as [tex]\(\frac{2}{3}\)[/tex], which translates to approximately 0.6667 or 66.67%.
2. The probability that any given day is a Friday, given that there are 7 days in a week, is [tex]\(\frac{1}{7}\)[/tex], which is approximately 0.1429 or 14.29%.
Step 2: Simulate the probabilities
### Available Methods for Simulation
- Method 1: Roll a die 8 times and count the number of times you roll a number greater than 2.
- Method 2: Pick a marble and replace it 100 times, counting the number of times you pick a yellow marble.
- Method 3: Roll a die, pick and replace a marble 100 times, and count the number of times you roll greater than 2 and pick a green marble.
- Method 4: Roll a die 8 times and count the number of times you roll a 2 and pick and replace a yellow marble.
- Method 5: Roll a die, pick and replace a marble 100 times, and count the number of times you roll a 5 and pick a green marble.
### Selecting the Appropriate Simulation
From these methods, we need to find the one that accurately simulates choosing a year at random and checking if the first day of winter will be both snowy and a Friday.
We need both probabilities to be correctly represented:
- Rolling a die to simulate the snow probability.
- Picking a marble from a hat to simulate the day being a Friday.
Correct Method:
- Roll a die, pick and replace a marble 100 times, and count the number of times you roll ≥ 5 and pick a green marble.
This correctly aligns as rolling ≥ 5 simulates the [tex]\(\frac{2}{3}\)[/tex] probability of snow (since rolling 5 or 6 out of 6 choices approximates to [tex]\(\frac{2}{3}\)[/tex]), and picking one specific color from 7 (green marble) simulates the [tex]\(\frac{1}{7}\)[/tex] probability of the day being a Friday.
Step 3: Finding the combined probability
From our given results:
1. Snow probability = 0.6667
2. Friday probability = 0.1429
To find the probability of both happening (snowy Friday):
[tex]\[ P(\text{Snowy Friday}) = \text{Snow probability} \times \text{Friday probability} \][/tex]
[tex]\[ P(\text{Snowy Friday}) = 0.6667 \times 0.1429 \approx 0.0952 \][/tex]
This result implies that there is approximately a 9.52% chance that the first day of winter will be a snowy Friday when choosing a year at random.
