4a. [2 marks]
The number of taxis arriving at Cardiff Central railway station can be modelled by a Poisson distribution.
During busy periods of the day, taxis arrive at a mean rate of 5.3 taxis every 10 minutes. Let T represent a random 10 minute busy period.
Find the probability that exactly 4 taxis arrive during T.
4b. [2 marks)
Find the most likely number of taxis that would arrive during T.
4c. (3 marks]
Given that more than 5 taxis arrive during T, find the probability that exactly 7 taxis arrive during T.
4d. [6 marks]
During quiet periods of the day, taxis arrive at a mean rate of 1.3 taxis every 10 minutes.
Find the probability that during a period of 15 minutes, of which the first 10 minutes is busy and the next 5 minutes is quiet, that exactly 2 taxis arrive


We will solve the questions one by one. 

### 4a. **Find the probability that exactly 4 taxis arrive during \( T \) (a 10-minute busy period).**

The number of taxis arriving during a 10-minute busy period follows a **Poisson distribution** with a mean rate of \( \lambda = 5.3 \) taxis per 10 minutes.

The formula for the probability mass function of a Poisson distribution is:
\[
P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}
\]
Where:
- \( \lambda = 5.3 \) (the mean number of taxis arriving in 10 minutes),
- \( k = 4 \) (the exact number of taxis we want to find the probability for).

Thus, the probability that exactly 4 taxis arrive is:
\[
P(X = 4) = \frac{5.3^4 e^{-5.3}}{4!}
\]

Let me calculate that.

### 4b. **Find the most likely number of taxis that would arrive during \( T \).**

For a Poisson distribution, the most likely number of taxis (the mode) is either \( \lfloor \lambda \rfloor \) or \( \lfloor \lambda \rfloor + 1 \), where \( \lambda \) is the mean rate.

Here, \( \lambda = 5.3 \), so the most likely number of taxis is either \( 5 \) or \( 6 \). We need to compare the probabilities for \( P(X = 5) \) and \( P(X = 6) \) to find the mode.

### 4c. **Given that more than 5 taxis arrive during \( T \), find the probability that exactly 7 taxis arrive during \( T \).**

We are asked to find \( P(X = 7 \mid X > 5) \). Using conditional probability:
\[
P(X = 7 \mid X > 5) = \frac{P(X = 7 \text{ and } X > 5)}{P(X > 5)} = \frac{P(X = 7)}{P(X > 5)}
\]

We already know the distribution is Poisson with \( \lambda = 5.3 \). Thus, we will calculate:
\[
P(X = 7) = \frac{5.3^7 e^{-5.3}}{7!}
\]
and
\[
P(X > 5) = 1 - P(X \leq 5) = 1 - \sum_{k=0}^{5} P(X = k)
\]

### 4d. **Find the probability that during a 15-minute period, exactly 2 taxis arrive, given the first 10 minutes are busy and the next 5 minutes are quiet.**

The total period is 15 minutes, consisting of:
- 10 busy minutes with mean arrival rate \( \lambda_1 = 5.3 \),
- 5 quiet minutes with mean arrival rate \( \lambda_2 = 1.3 \times \frac{5}{10} = 0.65 \).

The total number of taxis in the 15 minutes follows a Poisson distribution with mean:
\[
\lambda = \lambda_1 + \lambda_2 = 5.3 + 0.65 = 5.95
\]

Now, the probability of exactly 2 taxis arriving is:
\[
P(X = 2) = \frac{5.95^2 e^{-5.95}}{2!}
\]

I'll compute all the probabilities step by step now.Here are the results for each part:

### 4a. **Probability that exactly 4 taxis arrive during \( T \)**:
The probability is:
\[
P(X = 4) = 0.1641 \, (16.41\

The number of taxis arriving at Cardiff Central railway station can be modelled by a Poisson distribution. During busy periods of the day, taxis arrive at a mean rate of 5.3 taxis every 10 minutes. Let T represent a random 10 minute busy period. 4a. Find the probability that exactly 4 taxis arrive during T. 4b. Find the most likely number of taxis that would arrive during T. 4c. Given that more than 5 taxis arrive during T, find the probability that exactly 7 taxis arrive during T. 4d. During quiet periods of the day, taxis arrive at a mean rate of 1.3 taxis every 10 minutes. Find the probability that during a period of 15 minutes, of which the first 10 minutes is busy and the next 5 minutes is quiet, that exactly 2 taxis arrive.

Solve Poisson distribution problems on taxi arrivals at Cardiff Central Station. Find the probability of exactly 4 taxis arriving, the most likely number, conditional probability for 7 taxis, and calculate arrivals in a mixed busy-quiet period. Suitable for Grades 11-12 and college students.

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