STA hw4

Explain why 1.21 cannot be the probability of some event.

A probability must be between zero and one.

If you were using the relative frequency of an event to estimate the probability of the event, would it be better to use 100 trials or 500 trials? Explain.

It would be better to use 500 trials, because the law of large numbers would take effect.

If two events are mutually exclusive, can they occur concurrently? Explain.

No. By definition, mutually exclusive events cannot occur together.

What is the probability of the following. (a) An event A that is certain to occur? (b) An event B that is impossible?

(a) 1 (b) 0

Consider the following events for a driver selected at random from the general population. A = driver is under 25 years old B = driver has received a speeding ticket Translate each of the following phrases into symbols. (a) The probability the driver has received a speeding ticket and is under 25 years old. (b) The probability a driver who is under 25 years old has recieved a speeding ticket. (c) The probability a driver who has received a speeding ticket is 25 years old or older. (d) The probability the driver is under 25 years old or has received a speeding ticket. (e) The probability the driver has not received a speeding ticket or is under 25 years old.

(a) P(A and B) (b) P(B | A) (c) P(Ac | B) (d) P(A or B) (e) P(Bc or A)

Can the number 0.56 be the probability of an event? Explain.

Yes, it is a number between 0 and 1.

Explain why 120% cannot be the probability of some event.

A probability must be between zero and one.

Explain why −0.41 cannot be the probability of some event.

A probability must be between zero and one.

A national park is famous for its beautiful desert landscape and its many natural rock formations. The following table is based on information gathered by a park ranger of all rock formations of at least 3 feet. The height of the rock formation is rounded to the nearest foot. Height of rock formation, feet 3-9 10-29 30-49 50-74 75 and higher Number of rock formations in park 119 81 31 26 11 For a rock formation chosen at random from this park, use the preceding information to estimate the probability that the height of the rock formation is as follows. (Round your answers to two decimal places.) (a) 3 to 9 feet (b) 30 feet or taller (c) 3 to 49 feet (d) 10 to 74 feet (e) 75 feet or taller

(a) (b) (c) (d) (e)

You need to know the number of different arrangements possible for five distinct letters. You decide to use the permutations rule, but your friends tells you to use 5!. Who is correct? Explain.

Both methods are correct, since you are counting all possible arrangements of 5 items taken 5 at a time.

John runs a computer software store. Yesterday he counted 120 people who walked by the store, 54 of whom came into the store. Of the 54, only 26 bought something in the store. (a) Estimate the probability that a person who walks by the store will enter the store. (Round your answer to two decimal places.) (b) Estimate the probability that a person who walks into the store will buy something. (Round your answer to two decimal places.) (c) Estimate the probability that a person who walks by the store will come in and buy something. (Round your answer to two decimal places.) (d) Estimate the probability that a person who comes into the store will buy nothing. (Round your answer to two decimal places.)

(a) (b) (c) (d)

A botanist has developed a new hybrid cotton plant that can withstand insects better than other cotton plants. However, there is some concern about the germination of seeds from the new plant. To estimate the probability that a seed from the new plant will germinate, a random sample of 3000 seeds was planted in warm, moist soil. Of these seeds, 2000 germinated. (a) Use relative frequencies to estimate the probability that a seed will germinate. What is your estimate? (Enter your answer to 3 decimal places.) (b) Use relative frequencies to estimate the probability that a seed will not germinate. What is your estimate? (Enter your answer to 3 decimal places.) (c) Either a seed germinates or it does not. What is the sample space in this problem? Do the probabilities assigned to the sample space add up to 1? Should they add up to 1? Explain.

(a) 0.667 (b) 0.333 (c) germinate or not germinate Yes, because they cover the entire sample space. (d) no

(a) If you roll a single die and count the number of dots on top, what is the sample space of all possible outcomes? Are the outcomes equally likely? (b) Assign probabilities to the outcomes of the sample space of part (a). (Enter your answers as fractions.) Do the probabilities add up to 1? Should they add up to 1? Explain. (c) What is the probability of getting a number less than 2 on a single throw? (Enter your answer as a fraction.) (d) What is the probability of getting 1 or 2 on a single throw? (Enter your answer as a fraction.)

(a) 1, 2, 3, 4, 5, 6; equally likely (b) Outcome Probability 1 1/6 2 1/6 3 1/6 4 1/6 5 1/6 6 1/6 Yes, because these values cover the entire sample space. (c) (d) 1/3

You toss a pair of dice. (a) Determine the number of possible pairs of outcomes. (Recall that there are six possible outcomes for each die.) (b) There are three even numbers on each die. How many outcomes are possible with even numbers appearing on each die? (c) Probability extension: What is the probability that both dice will show an even number? (Use 2 decimal places.)

(a) 36 (b) 9 (c) 0.25

Consider a series of events. (a) How does a tree diagram help you list all the possible outcomes of a series of events? (b) How can you use a tree diagram to determine the total number of outcomes of a series of events?

(a) All possible outcomes are shown as distinct paths from the beginning of the diagram to the end of each last branch. (b) Counting the number of final branches gives the total number of outcomes.

Consider the following events for a college student selected at random. A = student is female B = student is majoring in business Translate each of the following phrases into symbols. (a) The probability the student is male or is majoring in business. (b) The probability a female student is majoring in business. (c) The probability a business major is female. (d) The probability the student is female and is not majoring in business (e) The probability the student is female and is majoring in business.

(a) P(Ac or B) (b) P(B | A) (c) P(A | B) (d) P(A and Bc) (e) P(A and B)

You roll two fair dice, one green and one red. (a) Are the outcomes on the dice independent? (b) Find P(1 on green die and 2 on red die). (Enter your answer as a fraction.) (c) Find P(2 on green die and 1 on red die). (Enter your answer as a fraction.) (d) Find P((1 on green die and 2 on red die) or (2 on green die and 1 on red die)). (Enter your answer as a fraction.)

(a) yes (b) 1/36 (c) 1/36 (d) 2/36

Consider the following. (a) Draw a tree diagram to display all the possible outcomes that can occur when you flip a coin and then toss a die. (b) How many outcomes contain a head and a number greater than 4? (c) Probability extension: Assuming the outcomes displayed in the tree diagram are all equally likely, what is the probability that you will get a head and a number greater than 4 when you flip a coin and toss a die? (Use 3 decimal places.)

(b) 2 (c) 1/6

What is the law of large numbers?

As the sample size increases, the relative frequency of outcomes gets closer to the theoretical probability of the outcome.

On a single toss of a fair coin, the probability of heads is 0.5 and the probability of tails is 0.5. If you toss a coin twice and get heads on the first toss, are you guaranteed to get tails on the second toss? Explain.

No, each outcome is equally likely regardless of the previous outcome.