Tetrahedral dice have four faces. Two fair tetrahedral dice, one red and one blue, have faces numbered 0, 1, 2, and 3 respectively. The dice are rolled and the numbers face down on the two dice are recorded. The random variable $R$ is the score on the red die and the random variable $B$ is the score on the blue die. (a) Find $P(R=3$ and $B=0)$. The random variable $T$ is $R$ multiplied by $B$. (b) Complete the diagram below to represent the sample space that shows all the possible values of $T$. (c) The table below represents the probability distribution of the random variable $T$: | $t$ | 0 | 1 | 2 | 3 | 4 | 6 | 9 | |----|---|---|---|---|---|---|---| | $P(T=t)$ | $a$ | $b$ | $1/8$ | $c$ | $1/8$ | $d$ | Find the values of $a, b, c$ and $d$. (d) Find the values of $E(T)$. (e) Var$(T)$.

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Tetrahedral dice have four faces - Edexcel - A-Level Maths Statistics - Question 7 - 2008 - Paper 1

Question 7

Tetrahedral dice have four faces. Two fair tetrahedral dice, one red and one blue, have faces numbered 0, 1, 2, and 3 respectively. The dice are rolled and the numbe... show full transcript

Worked Solution & Example Answer:Tetrahedral dice have four faces - Edexcel - A-Level Maths Statistics - Question 7 - 2008 - Paper 1

Step 1

Find $P(R=3$ and $B=0)$

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Answer

To find $P(R=3$ and $B=0)$ , we first note that the red die can land on one of four outcomes: 0, 1, 2, or 3, and likewise for the blue die. The total number of combinations when rolling both dice is 4 x 4 = 16.

The event where $R=3$ and $B=0$ happens only once (rolling a 3 on the red die and a 0 on the blue die), thus the probability is given by:

$P(R=3$ and $B=0) = \frac{1}{16}$

Step 2

Complete the diagram below to represent the sample space of T

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Answer

In this case, since $T = R \times B$ , we can fill out the values in the sample space diagram. The entries represent the product of the values of $R$ and $B$ :

3	6	9
2	2	3
0	0	0
$B$	0	1
$R$

Thus the completed diagram will look like this:

Step 3

The table represents the probability distribution of T

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Answer

To find $a$ , $b$ , $c$ , and $d$ , we note that since the total probability must equal 1: $a + b + \frac{1}{8} + c + \frac{1}{8} + d = 1$

We also know the outcomes for $T$ : 0 occurs $3$ times, 1 occurs $1$ time, 2 occurs $1$ time, 3 occurs $1$ time, 4 occurs $1$ time, 6 occurs $1$ time, and 9 occurs $1$ time. Therefore, we can set equations to find values of $a, b, c,$ and $d$ .

Through solving, we will find that:

$a = \frac{7}{16}$
$b = \frac{1}{16}$
$c = \frac{1}{8}$
$d = \frac{1}{16}$ .

Step 4

Find the values of E(T)

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Answer

To find the expected value, $E(T)$ , we use: $E(T) = \sum_{t} t \cdot P(T=t)$
Substituting the probabilities gives: $E(T) = 0 \cdot a + 1 \cdot b + 2 \cdot \frac{1}{8} + 3 \cdot c + 4 \cdot \frac{1}{8} + 6 \cdot d + 9 \cdot (\text{unneeded})$
This evaluates to $E(T) = 2.25$ .

Step 5

Find Var(T)

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Answer

To find the variance, we first compute $E(T^2)$ and then use: $Var(T) = E(T^2) - (E(T))^2$
Calculating $E(T^2)$ involves summing the probabilities of each outcome squared. Finally, plug in the expected value calculated before into the formula.

Tetrahedral dice have four faces - Edexcel - A-Level Maths Statistics - Question 7 - 2008 - Paper 1

Worked Solution & Example Answer:Tetrahedral dice have four faces - Edexcel - A-Level Maths Statistics - Question 7 - 2008 - Paper 1

Find $P(R=3$ and $B=0)$

Complete the diagram below to represent the sample space of T

The table represents the probability distribution of T

Find the values of E(T)

Find Var(T)

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