Natasha Awada, Paul La Rondie, Laurie Buchanan and, Jill Stevens Solutions for Chapter: Valid Comparisons and Informed Decisions: Probability Distributions, Exercise 22: Exercise 14F

A fair dice is rolled $40$ times. Find the expected number of sixes.

A card is drawn from a pack of $52$ cards, noted and returned. This is repeated until $40$ cards have been drawn. Find the expected number of hearts.

$X$ is a random variable such that $X~B\left(n,p\right)$. Given that the mean of the distribution is $10$, and $p=0.4$, find $n$.

A multiple choice test has $15$ questions, and each question has four possible answers. There is only one correct answer per question. If $X$ is "the number of questions a student guesses correctly", find: the distribution of $X$.

A multiple choice test has $15$ questions, and each question has four possible answers. There is only one correct answer per question. If $X$ is "the number of questions a student guesses correctly", find: the expectation of  $X$.

A multiple choice test has $15$ questions, and each question has four possible answers. There is only one correct answer per question. If $X$ is "the number of questions a student guesses correctly", find: the probability that, if a student answers the test purely by guessing, they will achieve the pass mark of $10$ or more.

A group of $100$ families, each with three children, are found to have the following number of girls.

$\begin{array}{|lcccc|}\hline \begin{array}{l}\text{ Number of }\\ \text{ girls }\end{array}& 0& 1& 2& 3\\ \text{ Frequency }& 13& 34& 40& 13\\ \hline\end{array}$

Using this sample of $100$ families to represent the population, estimate the probability that a new baby born is a girl.

A group of $100$ families, each with three children, are found to have the following number of girls.

$\begin{array}{|lcccc|}\hline \begin{array}{l}\text{ Number of }\\ \text{ girls }\end{array}& 0& 1& 2& 3\\ \text{ Frequency }& 13& 34& 40& 13\\ \hline\end{array}$

Another sample of $100$ families, each with three children, is taken from the same population. Calculate the number of families in this sample that you would expect to have two girls.