## MATH1005 Quizzes

Quiz 4: Probability
Question 1 Questions
The following ordered pairs of numbers $\left(x,y\right)$ were obtained in an experiment:
$\left(1,2\right),\left(2,4\right),\left(3,6\right),\left(3,7\right).$
Without performing calculations, what is the best description of the size of the correlation coefficient of the data ? Exactly one option must be correct)
 a) Slightly less than 1. b) Positive. c) Slightly greater than 2. d) Slightly greater than 1.

Choice (a) is correct!
By inspection, we see that the correlation coefficient must be positive but the magnitude of the correlation coefficient can never be greater than 1.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
We are told that in a random experiment there are five possible outcomes. Which of the following statements is true ? Exactly one option must be correct)
 a) If, after 20 trials, one outcome has not been observed then the probability that it will occur in the next trial is increased. b) If, after 20 trials, one outcome has been observed more often than the others then the probability that it will not occur in the next trial is increased. c) If, after 20 trials, one outcome has not been observed then the probability that it will occur in the next trial is unchanged. d) If the outcomes are equally likely then the trials are independent.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
In a random experiment the outcomes are independent so the first two statements are incorrect. We do not have enough information to comment on the probability of each outcome, so the fourth statement is incorrect.
Choice (d) is incorrect
A coin is tossed 6 times. What is the probability of exactly 2 heads occurring in the 6 tosses. Exactly one option must be correct)
 a) $\left(\begin{array}{c}\hfill 6\hfill \\ \hfill 2\hfill \end{array}\right){\left(\frac{1}{2}\right)}^{6}$ b) ${\left(\frac{1}{2}\right)}^{6}$ c) ${\left(\frac{1}{3}\right)}^{6}$ d) $\left(\begin{array}{c}\hfill 6\hfill \\ \hfill 2\hfill \end{array}\right){\left(\frac{1}{3}\right)}^{6}$

Choice (a) is correct!
The two heads can occur in any of the six tosses, so there are $\left(\begin{array}{c}\hfill 6\hfill \\ \hfill 2\hfill \end{array}\right)$ arrangements, with a probability of ${\left(\frac{1}{2}\right)}^{6}$ for each arrangement.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
A coin is tossed 6 times. What is the probability of getting 2 heads, followed by 4 tails ? Exactly one option must be correct)
 a) $\left(\begin{array}{c}\hfill 6\hfill \\ \hfill 2\hfill \end{array}\right){\left(\frac{1}{2}\right)}^{6}$ b) ${\left(\frac{1}{2}\right)}^{6}$ c) ${\left(\frac{1}{3}\right)}^{6}$ d) $\left(\begin{array}{c}\hfill 6\hfill \\ \hfill 2\hfill \end{array}\right){\left(\frac{1}{3}\right)}^{6}$

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
A coin is tossed and a fair six sided die is thrown. How many possible outcomes are there ? Exactly one option must be correct)
 a) 12 b) 6 c) 2 d) 8

Choice (a) is correct!
Each die face can come up with a head or a tail. The number of possible outcomes is therefore $6+6=12$.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Two fair dice are tossed. How many possible outcomes are there ? Exactly one option must be correct)
 a) 12 b) 6 c) 36 d) None of these.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
Two fair dice are thrown. What is the probability of the sum of 10 being obtained for the two uppermost faces ? Exactly one option must be correct)
 a) $\frac{1}{36}$ b) $\frac{1}{4}$ c) $\frac{1}{18}$ d) $\frac{1}{12}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
The possible pairs of faces are $\left(4,6\right)$, $\left(5,5\right)$ and $\left(6,4\right).$ That is, 3 out of a possible 36. The probability is therefore $\frac{3}{36}=\frac{1}{12}$.
What is the value of $\left(\begin{array}{c}\hfill n\hfill \\ \hfill r\hfill \end{array}\right)$ with $n=7$ and $r=2$ ? Exactly one option must be correct)
 a) 2520 b) 42 c) 21 d) 84

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
$\left(\begin{array}{c}\hfill n\hfill \\ \hfill r\hfill \end{array}\right)=\frac{n\phantom{\rule{0.3em}{0ex}}!}{r\phantom{\rule{0.3em}{0ex}}!\left(n-r\right)\phantom{\rule{0.3em}{0ex}}!}=\frac{7\phantom{\rule{0.3em}{0ex}}!}{2\phantom{\rule{0.3em}{0ex}}!\phantom{\rule{0.3em}{0ex}}5\phantom{\rule{0.3em}{0ex}}!}=\frac{7×6}{2}=21\phantom{\rule{0.3em}{0ex}}.$
Choice (d) is incorrect
A fair coin is tossed twice. What is the probability of obtaining a head and a tail ? Exactly one option must be correct)
 a) $\frac{1}{2}$ b) $\frac{1}{3}$ c) $\frac{1}{4}$ d) $\frac{1}{5}$

Choice (a) is correct!
The two possible outcomes are HT and TH, each of which has probability $\frac{1}{4}$. The probability of a head and a tail is therefore $\frac{1}{4}+\frac{1}{4}=\frac{1}{2}$.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
A coin is tossed until two heads occur on successive throws. The number of tosses is described by which of the following ? Exactly one option must be correct)
 a) It is a binomial distribution, $B\phantom{\rule{0.3em}{0ex}}\left(n,\frac{1}{2}\right)$. b) It is a binomial distribution, $B\phantom{\rule{0.3em}{0ex}}\left(n,\frac{2}{n}\right)$. c) It is not described by a binomial distribution. d) It is described by a geometric distribution.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect