1.8.1: Preparation N.8
- Page ID
- 147912
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)(1) Suppose a local food bank is selling raffle tickets as a fundraiser and claims that each raffle ticket has a 3% chance of winning one of several prizes. Additionally, suppose an animal shelter is also selling raffle tickets for a different fundraiser, and reveals that there are 1200 total tickets, and exactly 48 will win a prize.
(a) Which fundraiser offers a greater chance of winning a prize? Explain your answer using the likelihood of winning each raffle (in percentages).
(b) If you wanted to increase your chances of winning one of the raffles would it be better to buy five tickets or ten tickets? Explain your reasoning.
(c) If you bought 150 raffle tickets for the animal shelter’s fundraiser, how many tickets would you expect to be winners?
(d) If you bought 150 raffle tickets for the food bank’s fundraiser, how many tickets would you expect to be winners?
(e) If you bought 150 tickets for each raffle, is it possible that none of the tickets will win? Explain your answer.
(2) If there is a 61% chance that Carol gets accepted to Andover University, what is the chance, or probability, that Carol does not get accepted?
The probability of an event is defined as the likelihood that it will happen, and gives the chance that it will occur. Probability is often given as a percentage, which must always be between 0% (impossible) and 100% (certain). For instance, a team may be told they have a “30% chance of winning the game,” which would mean they are more likely to lose than win. Probability is sometimes described using words such as “unlikely,” “possible,” “even chance,” and “likely.” For example, if the probability of an event is 4%, we would say it is unlikely.
It is also common to represent probabilities as ratios. As an example, if you flip a coin, what is the likelihood that the coin will come up “heads”? The chances are 1 out of 2, or ½, since there is a head on only one of the two sides of a coin. You can think of the ratio as
\[probability = \dfrac{number\;of\;successful\;outcomes}{number\;of\;possible\;outcomes}\nonumber\]
This formula can be used to find any probability when all possible outcomes are equally likely. In the case of the coin, there was only 1 successful outcome (heads turning up) out of 2 possible outcomes (heads or tails).
(3) Suppose you roll a six-sided die.
(a) What is the probability that a 3 turns up?
(b) What is the probability that an even number turns up?
(c) What is the probability that a number greater than 2 turns up?
(4) Suppose that you have a bag containing 3 green marbles, 8 blue marbles and 13 red marbles. Suppose you reach into the bag and randomly draw one marble. You will win a treat if it is a green marble. What is the probability that you win a treat?
After Preparation N.8 (survey)
You should be able to do the following things for the next collaboration. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident).
Before beginning Collaboration N.8, you should understand the concepts and demonstrate the skills listed below:
| Skill or Concept: I can … | Rating from 1 to 5 |
| read information presented in a table. | |
| represent risks or probabilities by both percentages and ratios. |