Certainly! Let's go through the solution step-by-step:
1. Determine the Probability of Rain on the First Day of Spring:
From the historical data given, it is noted that rain fell on the first day of spring [tex]\(\frac{1}{3}\)[/tex] of the time. Therefore, the probability of rain on the first day of spring is:
[tex]\[ \text{P(Rain on the first day of spring)} = \frac{1}{3} \approx 0.3333 \][/tex]
2. Determine the Probability of the First Day of Spring Being Wednesday:
There are 7 days in a week, making each day equally likely to be the first day of spring. Therefore, the probability that the first day of spring falls on a Wednesday is:
[tex]\[ \text{P(Wednesday)} = \frac{1}{7} \approx 0.1429 \][/tex]
3. Calculate the Joint Probability of Both Events Happening:
To find the probability that the first day of spring is both rainy and falls on a Wednesday, we multiply the probabilities of the two independent events. Thus:
[tex]\[
\text{P(Rain on Wednesday)} = \text{P(Rain on the first day of spring)} \times \text{P(Wednesday)} = \frac{1}{3} \times \frac{1}{7}
\][/tex]
Substituting in the values, we get:
[tex]\[
\text{P(Rain on Wednesday)} = \left(\frac{1}{3}\right) \times \left(\frac{1}{7}\right) = \frac{1}{21} \approx 0.0476
\][/tex]
So, the probability that the first day of spring will be a rainy Wednesday is approximately [tex]\(0.0476\)[/tex], which can also be expressed as [tex]\(\frac{1}{21}\)[/tex] or approximately 4.76%.
