Math Test 3

Question content area top Part 1 In response to the​ question, "If a fair die is rolled​ twice, what is the probability of the event of rolling a pair of​ 5s?" a student​ replies, "One-third, because 16+16=13​." How do you​ respond?

. The student is not correct. The confusion probably lies in the fact that the student thinks that probabilities are additive. The student does not understand the Multiplication Rule for Probabilities. A partial tree diagram is 16→516 →5. ​Thus, P(5,5)=1616=136.

A class of 23 students is asked to find the probability of getting a sum that is a multiple of 3 when a pair of fair dice is rolled. The bar graph shows the different answers and their frequency. Compute the correct​ answer, and for each incorrect​ answer, explain the error made.

13​, ways to make a multiple of 3 sum

Question content area top Part 1 A student claims that if a fair coin is tossed and comes up heads 5 times in a​ row, then, according to the law of​ averages, the probability of tails on the next toss is greater than the probability of heads. What is your​ reply?

A fair coin has​ "no memory."​ Hence, the probability of the event of a tail on each toss is 12 regardless of how many tails appeared in previous tosses.

Consider the experiment of drawing a single card from a standard deck of 52 cards and determine which are uniform sample​ spaces, that​ is, sample spaces with equally likely outcomes. a. ​{face card, not face​ card} b.​ {club, diamond,​ heart, spade} c.​ {black, red} d. ​{king, queen,​ jack, ace,​ even-numbered card,​ odd-numbered card}

A) ​{face card, not face​ card} is not a uniform sample space because this sample space does not have equally likely outcomes. B)​{club, diamond,​ heart, spade} is a uniform sample space because this sample space has equally likely outcomes. C) ​{black, red} is a uniform sample space because this sample space has equally likely outcomes. D)​{king, queen,​ jack, ace,​ even-numbered card,​ odd-numbered card} is not a uniform sample space because this sample space does not have equally likely outcomes.

For each of the following​ letters, describe an​ event, if​ possible, that has the approximate probability marked by the letter on the probability line.

A)Event A is an impossible event such as rolling a 10 on a single roll of a standard die. B)Event B has a low​ probability, such as the chance of rain being​ 20%. C)Event C is around​ 0.5, such as obtaining a head when tossing a fair coin. D)Event D has a high probability such as tossing a number less than 6 on a standard die. E)Event E has a probability of​ 1, such as tossing a head or a tail on a fair coin. F)Event F has a probability greater than​ 1, which cannot happen.

Question content area top Part 1 If events A and B are from the same sample​ space, and if ​P(A)=0.8 and ​P(B)=​0.9, can events A and B be mutually​ exclusive? Explain.

All probabilities are less than or equal to 1 and greater than or equal to 0. If events are mutually​ exclusive, then​ P(A ∪ ​B)=​P(A)+​P(B). ​Therefore, if A and B are mutually​ exclusive, P(A ∪ ​B)=​P(A)+​P(B)=0.8+0.9=​1.7, which is impossible.​ Therefore, events A and B are not mutually exclusive.

There are two bags each containing red balls and yellow balls. Bag A contains 1 red and 4 yellow balls. Bag B contains 3 red and 13 yellow balls. Alva says that you should always choose bag B if you are trying to draw a red ball because it has more red balls than bag A. How do you​ respond?

Alva's reasoning is not correct. If she chooses bag​ A, her chances of getting a red ball are 15​, which is greater than her chances of getting a red ball if she chooses bag​ B, 316.

Bobbie says that when she shoots a free throw in​ basketball, she will either make it or miss it. Because there are only two outcomes and one of them is making a​ basket, Bobbie claims the probability of her making a free throw is 12. Explain whether​ Bobbie's reasoning is correct.

Bobbie's reasoning is not correct. The outcomes of making a basket or missing it are probably not equally likely.

How might we use a​ random-digit table to simulate each of the​ following? a. Tossing a single die b. Choosing three people at random from a group of 20 people c. Spinning the​ spinner, where the probability of each color is as shown in the figure to the right

Let​ 1, 2,​ 3, 4,​ 5, and 6 represent the numbers on the die and ignore the numbers​ 0, 7,​ 8, 9. Number the persons​ 01, 02,...,​ 19, 20. Go to the​ random-digit table and mark off groups of two. The three persons chosen are the first three whose numbers appear. Represent red by the numbers​ 0, 1,​ 2, 3,​ 4; green by the numbers​ 5, 6,​ 7; yellow by the number​ 8; and white by the number 9.

Could we use a thumb tack to simulate the birth of boys and​ girls? Why or why​ not?

Not work not equally

How can the faces of two cubes be numbered so that when they are​ rolled, the resulting sum is a number 1 to 12 inclusive and each sum has the same​ probability?

Number them so that one cube has the numbers​ 0, 0,​ 0, 3,​ 3, 3, and the other cube has the numbers​ 1, 2,​ 3, 7,​ 8, 9.

Board games that use dice sometimes give children the wrong impression of the probability of rolling a certain number. For​ example, children often think that rolling a 6 is less likely than rolling a 3 or a 4. Describe an experiment that will help children understand that the probability of rolling each number on a die is equally likely.

Rolling a die 100 times manually or using technology.

Ian and Sophia flip a fair coin ten times and record their​ results; they find that the coin landed on heads eight times. Ian​ says, "This means the coin will land on heads​ 80% of the​ time!" Sophia​ says, "I​ don't think​ so; if we flip it many more​ times, our results should be closer to it landing on heads about​ 50% of the​ time." Who is​ right? How do you​ know?

Sophia is​ right, because of the Law of Large Numbers​ (Bernoulli's Theorem).

Question content area top Part 1 You have the choice of flipping a fair coin and winning a game when heads​ appear, or you can roll a single standard die and you win if an even number appears. Explain why the choice of coin or die does not matter in regard to your winning.

The choice does not matter because both winning events have probability 12.

Question content area top Part 1 Make up a game in which the players have an equal chance of winning and the game involves rolling two regular dice.

The game could be player A wins if the sum of the numbers is even and player B wins if the sum of the numbers is odd.

Explain the error made in calculating 118.

The probability 118 was found using the ways to sum to 3.

Explain the error made in calculating 19.

The probability 19 was found using the multiples of 3 divided by the total number of elements in the sample space.

Explain the error made in calculating 411.

The probability 411 was found by dividing the number of multiples of 3 by the number of different sums that can be obtained.

A student claims that​ "if the probability of an event is 35​, then there are three ways the event can occur and only five elements in the sample​ space." How do you​ respond?

The probability of an event is a ratio and does not necessarily reflect the number of elements in the event or in the sample space.

An experiment consists of tossing a fair coin twice. The student reasons that there are three possible​ outcomes: two​ heads, one head and one​ tail, or two tails.​ Thus, ​P(HH)=13. What is your​ reply?

The student is wrong because although there are 3 different​ outcomes, the outcomes are not equally likely. There are two ways get the outcome of one head and one tail.​ Thus, ​P(HH)=14.

A student observes the spinner to the right and claims that the color red has the highest probability of​ appearing, since there are two red areas on the spinner. What is your​ reply?

The student is wrong since the area of the red regions is smaller than the area of the green​ region, so the color red does not have the highest probability of appearing.

Explain the error made in calculating 13.

There was no error made and 13 is correct.

A student would like to know the difference between two events being independent and two events being mutually exclusive. How would you answer​ her?

Two events are independent when the occurrence or nonoccurrence of one event has no influence on the outcome of the other event. Two events are mutually exclusive when they have no outcomes in common.

Question content area top left Part 1 Zoe is playing a game in which she draws one ball from one of the boxes shown. She wins if she draws a white ball from either box​ #1 or box​ #2. She says that in order to maximize her chances of winning she will always pick box​ #2 because it has more white balls. Is she​ correct? Why?

Zoe is incorrect. Picking box​ #1 will maximize her chances of​ winning, because it has the higher probability of picking a white balls.

Let the universal​ set, U, be the set of students at Central​ High, A be the set of students taking​ algebra, and C be the set of students taking chemistry. If a student is selected at​ random, describe in words what is meant by each of the following probabilities. a. P(A ∪ C) b. P(A ∩ C) c. 1−​P(C)

a. Describe the meaning of the probability P(A ∪ C) in words. Choose the correct answer below. A. The probability of a student taking algebra or chemistry b. Describe the meaning of the probability P(A ∩ C) in words. Choose the correct answer below. C. The probability of a student taking algebra and chemistry c. Describe the meaning of the probability 1−​P(C) in words. Choose the correct answer below. prob not taking chemistry

Suppose the figure to the right is a dartboard with no spinner. Would you expect each color space to have the same probability of being hit with a dart on a random​ throw? Explain your answer.

​No, because the areas of the different sectors of the dartboard are​ unequal, we would not expect the probabilities to be equal.