Two fair dice are thrown. Find the probability of obtaining a sum of 7 or 11.

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6. Probability of obtaining a sum of 7 or 11

Determine the total number of possible outcomes when two fair dice are thrown.
Each die has 6 faces, so the total number of outcomes is $6 \times 6 = 36$ .
List the outcomes that result in a sum of 7.
The pairs are $(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)$ .
There are 6 outcomes that sum to 7.
The probability of getting a sum of 7 is $P(\text{sum of 7}) = \frac{6}{36}$ .
List the outcomes that result in a sum of 11.
The pairs are $(5,6), (6,5)$ .
There are 2 outcomes that sum to 11.
The probability of getting a sum of 11 is $P(\text{sum of 11}) = \frac{2}{36}$ .
Since the events are mutually exclusive (a sum cannot be both 7 and 11 simultaneously), add their probabilities to find the probability of obtaining a sum of 7 or 11.
$P(\text{sum of 7 or 11}) = P(\text{sum of 7}) + P(\text{sum of 11})$ $P(\text{sum of 7 or 11}) = \frac{6}{36} + \frac{2}{36}$ $P(\text{sum of 7 or 11}) = \frac{8}{36}$
Simplify the probability.
$P(\text{sum of 7 or 11}) = \frac{2}{9}$
The probability of obtaining a sum of 7 or 11 is $\boxed{\frac{2}{9}}$ .

7. (a) Find the position vector of P
Identify the given position vectors and the ratio.
The position vector of point F is $\vec{f} = 3\mathbf{i} - 2\mathbf{j}$ .
The position vector of point G is $\vec{g} = \mathbf{i} + \mathbf{j}$ .
Point P divides FG in the ratio $5:3$ . This means $m=5$ and $n=3$ .
Use the section formula for internal division.
If a point P divides the line segment joining points F and G with position vectors $\vec{f}$ and $\vec{g}$ respectively, in the ratio $m:n$ , then the position vector of P, $\vec{p}$ , is given by:
$\vec{p} = \frac{n\vec{f} + m\vec{g}}{m+n}$
Substitute the given values into the formula.
$\vec{p} = \frac{3(3\mathbf{i} - 2\mathbf{j}) + 5(\mathbf{i} + \mathbf{j})}{5+3}$
Simplify the expression.
$\vec{p} = \frac{(9\mathbf{i} - 6\mathbf{j}) + (5\mathbf{i} + 5\mathbf{j})}{8}$ $\vec{p} = \frac{(9+5)\mathbf{i} + (-6+5)\mathbf{j}}{8}$ $\vec{p} = \frac{14\mathbf{i} - 1\mathbf{j}}{8}$ $\vec{p} = \frac{14}{8}\mathbf{i} - \frac{1}{8}\mathbf{j}$ $\vec{p} = \frac{7}{4}\mathbf{i} - \frac{1}{8}\mathbf{j}$
The position vector of P is $\boxed{\frac{7}{4}\mathbf{i} - \frac{1}{8}\mathbf{j}}$ .

Two fair dice are thrown. Find the probability of obtaining a sum of 7 or 11.

Step-by-step solution

Determine the total number of possible outcomes when two fair dice are thrown.

List the outcomes that result in a sum of 7.

List the outcomes that result in a sum of 11.

Since the events are mutually exclusive (a sum cannot be both 7 and 11 simultaneously), add their probabilities to find the probability of obtaining a sum of 7 or 11.

Simplify the probability.

Identify the given position vectors and the ratio.

Use the section formula for internal division.

Substitute the given values into the formula.

Simplify the expression.

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