Example 2:
(a) Find the expectation of the number on a die when thrown.
(b) Two unbiased dice are thrown. Find the expected values of the sum of numbers of points
on them.

Question

Example 2: (a) Find the expectation of the number on a die when thrown. (b) Two unbiased dice are thrown. Find the expected values of the sum of numbers of points on them.

UpStudy AI · Accepted Answer

For part (a), we need to find the expectation of the number on a die when thrown.

The expectation of a random variable is calculated by taking the sum of all possible values of the random variable multiplied by their respective probabilities.

For a fair six-sided die, the possible values are 1, 2, 3, 4, 5, and 6, each with a probability of 1/6.

The expectation of the number on a die when thrown is calculated as follows:
$$ E(X) = \sum_{i=1}^{6} x_i \cdot P(x_i) $$
where $$ x_i $$ is the value of the die and $$ P(x_i) $$ is the probability of getting that value.

Let's calculate the expectation for a single die throw.
Calculate the value by following steps:
- step0: Calculate:
  $$1\times \frac{1}{6}+2\times \frac{1}{6}+3\times \frac{1}{6}+4\times \frac{1}{6}+5\times \frac{1}{6}+6\times \frac{1}{6}$$
- step1: Multiply:
  $$\frac{1}{6}+2\times \frac{1}{6}+3\times \frac{1}{6}+4\times \frac{1}{6}+5\times \frac{1}{6}+6\times \frac{1}{6}$$
- step2: Reduce the numbers:
  $$\frac{1}{6}+\frac{1}{3}+3\times \frac{1}{6}+4\times \frac{1}{6}+5\times \frac{1}{6}+6\times \frac{1}{6}$$
- step3: Reduce the numbers:
  $$\frac{1}{6}+\frac{1}{3}+\frac{1}{2}+4\times \frac{1}{6}+5\times \frac{1}{6}+6\times \frac{1}{6}$$
- step4: Multiply the numbers:
  $$\frac{1}{6}+\frac{1}{3}+\frac{1}{2}+\frac{2}{3}+5\times \frac{1}{6}+6\times \frac{1}{6}$$
- step5: Multiply:
  $$\frac{1}{6}+\frac{1}{3}+\frac{1}{2}+\frac{2}{3}+\frac{5}{6}+6\times \frac{1}{6}$$
- step6: Reduce the numbers:
  $$\frac{1}{6}+\frac{1}{3}+\frac{1}{2}+\frac{2}{3}+\frac{5}{6}+1$$
- step7: Reduce fractions to a common denominator:
  $$\frac{1}{6}+\frac{2}{3\times 2}+\frac{3}{2\times 3}+\frac{2\times 2}{3\times 2}+\frac{5}{6}+\frac{3\times 2}{3\times 2}$$
- step8: Multiply the numbers:
  $$\frac{1}{6}+\frac{2}{6}+\frac{3}{2\times 3}+\frac{2\times 2}{3\times 2}+\frac{5}{6}+\frac{3\times 2}{3\times 2}$$
- step9: Multiply the numbers:
  $$\frac{1}{6}+\frac{2}{6}+\frac{3}{6}+\frac{2\times 2}{3\times 2}+\frac{5}{6}+\frac{3\times 2}{3\times 2}$$
- step10: Multiply the numbers:
  $$\frac{1}{6}+\frac{2}{6}+\frac{3}{6}+\frac{2\times 2}{6}+\frac{5}{6}+\frac{3\times 2}{3\times 2}$$
- step11: Multiply the numbers:
  $$\frac{1}{6}+\frac{2}{6}+\frac{3}{6}+\frac{2\times 2}{6}+\frac{5}{6}+\frac{3\times 2}{6}$$
- step12: Transform the expression:
  $$\frac{1+2+3+2\times 2+5+3\times 2}{6}$$
- step13: Multiply the numbers:
  $$\frac{1+2+3+4+5+3\times 2}{6}$$
- step14: Multiply the numbers:
  $$\frac{1+2+3+4+5+6}{6}$$
- step15: Add the numbers:
  $$\frac{21}{6}$$
- step16: Reduce the fraction:
  $$\frac{7}{2}$$
The expectation of the number on a die when thrown is 3.5.

For part (b), we need to find the expected values of the sum of numbers of points on two unbiased dice.

When two unbiased dice are thrown, the possible sums of the numbers on the dice range from 2 to 12.

The expected value of the sum of numbers on two dice is calculated by taking the sum of all possible sums multiplied by their respective probabilities.

Let's calculate the expected value of the sum of numbers on two dice.
Calculate the value by following steps:
- step0: Calculate:
  $$2\times \frac{1}{36}+3\left(\frac{2}{36}\right)+4\left(\frac{3}{36}\right)+5\left(\frac{4}{36}\right)+6\times \frac{5}{36}+7\left(\frac{6}{36}\right)+8\times \frac{5}{36}+9\left(\frac{4}{36}\right)+10\left(\frac{3}{36}\right)+11\left(\frac{2}{36}\right)+12\times \frac{1}{36}$$
- step1: Reduce the fraction:
  $$2\times \frac{1}{36}+3\times \frac{1}{18}+4\left(\frac{3}{36}\right)+5\left(\frac{4}{36}\right)+6\times \frac{5}{36}+7\left(\frac{6}{36}\right)+8\times \frac{5}{36}+9\left(\frac{4}{36}\right)+10\left(\frac{3}{36}\right)+11\left(\frac{2}{36}\right)+12\times \frac{1}{36}$$
- step2: Reduce the fraction:
  $$2\times \frac{1}{36}+3\times \frac{1}{18}+4\times \frac{1}{12}+5\left(\frac{4}{36}\right)+6\times \frac{5}{36}+7\left(\frac{6}{36}\right)+8\times \frac{5}{36}+9\left(\frac{4}{36}\right)+10\left(\frac{3}{36}\right)+11\left(\frac{2}{36}\right)+12\times \frac{1}{36}$$
- step3: Reduce the fraction:
  $$2\times \frac{1}{36}+3\times \frac{1}{18}+4\times \frac{1}{12}+5\times \frac{1}{9}+6\times \frac{5}{36}+7\left(\frac{6}{36}\right)+8\times \frac{5}{36}+9\left(\frac{4}{36}\right)+10\left(\frac{3}{36}\right)+11\left(\frac{2}{36}\right)+12\times \frac{1}{36}$$
- step4: Reduce the fraction:
  $$2\times \frac{1}{36}+3\times \frac{1}{18}+4\times \frac{1}{12}+5\times \frac{1}{9}+6\times \frac{5}{36}+7\times \frac{1}{6}+8\times \frac{5}{36}+9\left(\frac{4}{36}\right)+10\left(\frac{3}{36}\right)+11\left(\frac{2}{36}\right)+12\times \frac{1}{36}$$
- step5: Reduce the fraction:
  $$2\times \frac{1}{36}+3\times \frac{1}{18}+4\times \frac{1}{12}+5\times \frac{1}{9}+6\times \frac{5}{36}+7\times \frac{1}{6}+8\times \frac{5}{36}+9\times \frac{1}{9}+10\left(\frac{3}{36}\right)+11\left(\frac{2}{36}\right)+12\times \frac{1}{36}$$
- step6: Reduce the fraction:
  $$2\times \frac{1}{36}+3\times \frac{1}{18}+4\times \frac{1}{12}+5\times \frac{1}{9}+6\times \frac{5}{36}+7\times \frac{1}{6}+8\times \frac{5}{36}+9\times \frac{1}{9}+10\times \frac{1}{12}+11\left(\frac{2}{36}\right)+12\times \frac{1}{36}$$
- step7: Reduce the fraction:
  $$2\times \frac{1}{36}+3\times \frac{1}{18}+4\times \frac{1}{12}+5\times \frac{1}{9}+6\times \frac{5}{36}+7\times \frac{1}{6}+8\times \frac{5}{36}+9\times \frac{1}{9}+10\times \frac{1}{12}+11\times \frac{1}{18}+12\times \frac{1}{36}$$
- step8: Reduce the numbers:
  $$\frac{1}{18}+3\times \frac{1}{18}+4\times \frac{1}{12}+5\times \frac{1}{9}+6\times \frac{5}{36}+7\times \frac{1}{6}+8\times \frac{5}{36}+9\times \frac{1}{9}+10\times \frac{1}{12}+11\times \frac{1}{18}+12\times \frac{1}{36}$$
- step9: Reduce the numbers:
  $$\frac{1}{18}+\frac{1}{6}+4\times \frac{1}{12}+5\times \frac{1}{9}+6\times \frac{5}{36}+7\times \frac{1}{6}+8\times \frac{5}{36}+9\times \frac{1}{9}+10\times \frac{1}{12}+11\times \frac{1}{18}+12\times \frac{1}{36}$$
- step10: Reduce the numbers:
  $$\frac{1}{18}+\frac{1}{6}+\frac{1}{3}+5\times \frac{1}{9}+6\times \frac{5}{36}+7\times \frac{1}{6}+8\times \frac{5}{36}+9\times \frac{1}{9}+10\times \frac{1}{12}+11\times \frac{1}{18}+12\times \frac{1}{36}$$
- step11: Multiply:
  $$\frac{1}{18}+\frac{1}{6}+\frac{1}{3}+\frac{5}{9}+6\times \frac{5}{36}+7\times \frac{1}{6}+8\times \frac{5}{36}+9\times \frac{1}{9}+10\times \frac{1}{12}+11\times \frac{1}{18}+12\times \frac{1}{36}$$
- step12: Reduce the numbers:
  $$\frac{1}{18}+\frac{1}{6}+\frac{1}{3}+\frac{5}{9}+\frac{5}{6}+7\times \frac{1}{6}+8\times \frac{5}{36}+9\times \frac{1}{9}+10\times \frac{1}{12}+11\times \frac{1}{18}+12\times \frac{1}{36}$$
- step13: Multiply:
  $$\frac{1}{18}+\frac{1}{6}+\frac{1}{3}+\frac{5}{9}+\frac{5}{6}+\frac{7}{6}+8\times \frac{5}{36}+9\times \frac{1}{9}+10\times \frac{1}{12}+11\times \frac{1}{18}+12\times \frac{1}{36}$$
- step14: Multiply the numbers:
  $$\frac{1}{18}+\frac{1}{6}+\frac{1}{3}+\frac{5}{9}+\frac{5}{6}+\frac{7}{6}+\frac{10}{9}+9\times \frac{1}{9}+10\times \frac{1}{12}+11\times \frac{1}{18}+12\times \frac{1}{36}$$
- step15: Reduce the numbers:
  $$\frac{1}{18}+\frac{1}{6}+\frac{1}{3}+\frac{5}{9}+\frac{5}{6}+\frac{7}{6}+\frac{10}{9}+1+10\times \frac{1}{12}+11\times \frac{1}{18}+12\times \frac{1}{36}$$
- step16: Multiply the numbers:
  $$\frac{1}{18}+\frac{1}{6}+\frac{1}{3}+\frac{5}{9}+\frac{5}{6}+\frac{7}{6}+\frac{10}{9}+1+\frac{5}{6}+11\times \frac{1}{18}+12\times \frac{1}{36}$$
- step17: Multiply:
  $$\frac{1}{18}+\frac{1}{6}+\frac{1}{3}+\frac{5}{9}+\frac{5}{6}+\frac{7}{6}+\frac{10}{9}+1+\frac{5}{6}+\frac{11}{18}+12\times \frac{1}{36}$$
- step18: Reduce the numbers:
  $$\frac{1}{18}+\frac{1}{6}+\frac{1}{3}+\frac{5}{9}+\frac{5}{6}+\frac{7}{6}+\frac{10}{9}+1+\frac{5}{6}+\frac{11}{18}+\frac{1}{3}$$
- step19: Reduce fractions to a common denominator:
  $$\frac{1}{18}+\frac{3}{6\times 3}+\frac{6}{3\times 6}+\frac{5\times 2}{9\times 2}+\frac{5\times 3}{6\times 3}+\frac{7\times 3}{6\times 3}+\frac{10\times 2}{9\times 2}+\frac{9\times 2}{9\times 2}+\frac{5\times 3}{6\times 3}+\frac{11}{18}+\frac{6}{3\times 6}$$
- step20: Multiply the numbers:
  $$\frac{1}{18}+\frac{3}{18}+\frac{6}{3\times 6}+\frac{5\times 2}{9\times 2}+\frac{5\times 3}{6\times 3}+\frac{7\times 3}{6\times 3}+\frac{10\times 2}{9\times 2}+\frac{9\times 2}{9\times 2}+\frac{5\times 3}{6\times 3}+\frac{11}{18}+\frac{6}{3\times 6}$$
- step21: Multiply the numbers:
  $$\frac{1}{18}+\frac{3}{18}+\frac{6}{18}+\frac{5\times 2}{9\times 2}+\frac{5\times 3}{6\times 3}+\frac{7\times 3}{6\times 3}+\frac{10\times 2}{9\times 2}+\frac{9\times 2}{9\times 2}+\frac{5\times 3}{6\times 3}+\frac{11}{18}+\frac{6}{3\times 6}$$
- step22: Multiply the numbers:
  $$\frac{1}{18}+\frac{3}{18}+\frac{6}{18}+\frac{5\times 2}{18}+\frac{5\times 3}{6\times 3}+\frac{7\times 3}{6\times 3}+\frac{10\times 2}{9\times 2}+\frac{9\times 2}{9\times 2}+\frac{5\times 3}{6\times 3}+\frac{11}{18}+\frac{6}{3\times 6}$$
- step23: Multiply the numbers:
  $$\frac{1}{18}+\frac{3}{18}+\frac{6}{18}+\frac{5\times 2}{18}+\frac{5\times 3}{18}+\frac{7\times 3}{6\times 3}+\frac{10\times 2}{9\times 2}+\frac{9\times 2}{9\times 2}+\frac{5\times 3}{6\times 3}+\frac{11}{18}+\frac{6}{3\times 6}$$
- step24: Multiply the numbers:
  $$\frac{1}{18}+\frac{3}{18}+\frac{6}{18}+\frac{5\times 2}{18}+\frac{5\times 3}{18}+\frac{7\times 3}{18}+\frac{10\times 2}{9\times 2}+\frac{9\times 2}{9\times 2}+\frac{5\times 3}{6\times 3}+\frac{11}{18}+\frac{6}{3\times 6}$$
- step25: Multiply the numbers:
  $$\frac{1}{18}+\frac{3}{18}+\frac{6}{18}+\frac{5\times 2}{18}+\frac{5\times 3}{18}+\frac{7\times 3}{18}+\frac{10\times 2}{18}+\frac{9\times 2}{9\times 2}+\frac{5\times 3}{6\times 3}+\frac{11}{18}+\frac{6}{3\times 6}$$
- step26: Multiply the numbers:
  $$\frac{1}{18}+\frac{3}{18}+\frac{6}{18}+\frac{5\times 2}{18}+\frac{5\times 3}{18}+\frac{7\times 3}{18}+\frac{10\times 2}{18}+\frac{9\times 2}{18}+\frac{5\times 3}{6\times 3}+\frac{11}{18}+\frac{6}{3\times 6}$$
- step27: Multiply the numbers:
  $$\frac{1}{18}+\frac{3}{18}+\frac{6}{18}+\frac{5\times 2}{18}+\frac{5\times 3}{18}+\frac{7\times 3}{18}+\frac{10\times 2}{18}+\frac{9\times 2}{18}+\frac{5\times 3}{18}+\frac{11}{18}+\frac{6}{3\times 6}$$
- step28: Multiply the numbers:
  $$\frac{1}{18}+\frac{3}{18}+\frac{6}{18}+\frac{5\times 2}{18}+\frac{5\times 3}{18}+\frac{7\times 3}{18}+\frac{10\times 2}{18}+\frac{9\times 2}{18}+\frac{5\times 3}{18}+\frac{11}{18}+\frac{6}{18}$$
- step29: Transform the expression:
  $$\frac{1+3+6+5\times 2+5\times 3+7\times 3+10\times 2+9\times 2+5\times 3+11+6}{18}$$
- step30: Multiply the numbers:
  $$\frac{1+3+6+10+5\times 3+7\times 3+10\times 2+9\times 2+5\times 3+11+6}{18}$$
- step31: Multiply the numbers:
  $$\frac{1+3+6+10+15+7\times 3+10\times 2+9\times 2+5\times 3+11+6}{18}$$
- step32: Multiply the numbers:
  $$\frac{1+3+6+10+15+21+10\times 2+9\times 2+5\times 3+11+6}{18}$$
- step33: Multiply the numbers:
  $$\frac{1+3+6+10+15+21+20+9\times 2+5\times 3+11+6}{18}$$
- step34: Multiply the numbers:
  $$\frac{1+3+6+10+15+21+20+18+5\times 3+11+6}{18}$$
- step35: Multiply the numbers:
  $$\frac{1+3+6+10+15+21+20+18+15+11+6}{18}$$
- step36: Add the numbers:
  $$\frac{126}{18}$$
- step37: Divide the terms:
  $$7$$
The expected value of the sum of numbers on two unbiased dice is 7.
(a) The expectation of a single die throw is 3.5. (b) The expected sum of two dice is 7.

Example 2:
(a) Find the expectation of the number on a die when thrown.
(b) Two unbiased dice are thrown. Find the expected values of the sum of numbers of points
on them.

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Probability Questions