A die is rolled what is the probability of getting an even number or a number greater than 4 brainly

Figure 3.3 Sample Spaces and Probability

Since the whole sample space S is an event that is certain to occur, the sum of the probabilities of all the outcomes must be the number 1.

In ordinary language probabilities are frequently expressed as percentages. For example, we would say that there is a 70% chance of rain tomorrow, meaning that the probability of rain is 0.70. We will use this practice here, but in all the computational formulas that follow we will use the form 0.70 and not 70%.

A coin is called “balanced” or “fair” if each side is equally likely to land up. Assign a probability to each outcome in the sample space for the experiment that consists of tossing a single fair coin.

Solution:

With the outcomes labeled h for heads and t for tails, the sample space is the set S={h,t}. Since the outcomes have the same probabilities, which must add up to 1, each outcome is assigned probability 1/2.

A die is called “balanced” or “fair” if each side is equally likely to land on top. Assign a probability to each outcome in the sample space for the experiment that consists of tossing a single fair die. Find the probabilities of the events E: “an even number is rolled” and T: “a number greater than two is rolled.”

Solution:

With outcomes labeled according to the number of dots on the top face of the die, the sample space is the set S={1,2,3,4,5,6}. Since there are six equally likely outcomes, which must add up to 1, each is assigned probability 1/6.

Since E={2,4,6}, P(E)=1∕6+1∕6+1∕6=3∕6=1∕2.

Since T={3,4,5,6}, P(T)=4∕6=2∕3.

Two fair coins are tossed. Find the probability that the coins match, i.e., either both land heads or both land tails.

Solution:

In Note 3.8 "Example 3" we constructed the sample space S={2h,2t,d} for the situation in which the coins are identical and the sample space S′={hh,ht,th,tt} for the situation in which the two coins can be told apart.

The theory of probability does not tell us how to assign probabilities to the outcomes, only what to do with them once they are assigned. Specifically, using sample space S, matching coins is the event M={2h,2t}, which has probability P(2h)+P(2t). Using sample space S′, matching coins is the event M′={hh,tt}, which has probability P(hh)+P(tt). In the physical world it should make no difference whether the coins are identical or not, and so we would like to assign probabilities to the outcomes so that the numbers P(M) and P(M′) are the same and best match what we observe when actual physical experiments are performed with coins that seem to be fair. Actual experience suggests that the outcomes in S′ are equally likely, so we assign to each probability 1∕4, and then

Similarly, from experience appropriate choices for the outcomes in S are:

which give the same final answer

The previous three examples illustrate how probabilities can be computed simply by counting when the sample space consists of a finite number of equally likely outcomes. In some situations the individual outcomes of any sample space that represents the experiment are unavoidably unequally likely, in which case probabilities cannot be computed merely by counting, but the computational formula given in the definition of the probability of an event must be used.

The breakdown of the student body in a local high school according to race and ethnicity is 51% white, 27% black, 11% Hispanic, 6% Asian, and 5% for all others. A student is randomly selected from this high school. (To select “randomly” means that every student has the same chance of being selected.) Find the probabilities of the following events:

1. B: the student is black,
2. M: the student is minority (that is, not white),
3. N: the student is not black.

Solution:

The experiment is the action of randomly selecting a student from the student population of the high school. An obvious sample space is S={w,b,h,a,o}. Since 51% of the students are white and all students have the same chance of being selected, P(w)=0.51, and similarly for the other outcomes. This information is summarized in the following table:

1. Since B={b}, P(B)=P(b)=0.27.
2. Since M={b,h,a,o}, P(M)=P(b)+P(h)+P(a)+P(o)=0.27+0.11+0.06+0.05=0.49
3. Since N={w,h,a,o}, P(N)=P(w)+P(h)+P(a)+P(o)=0.51+0.11+0.06+0.05=0.73

The student body in the high school considered in Note 3.18 "Example 8" may be broken down into ten categories as follows: 25% white male, 26% white female, 12% black male, 15% black female, 6% Hispanic male, 5% Hispanic female, 3% Asian male, 3% Asian female, 1% male of other minorities combined, and 4% female of other minorities combined. A student is randomly selected from this high school. Find the probabilities of the following events:

1. B: the student is black,
2. MF: the student is minority female,
3. FN: the student is female and is not black.

Solution:

Now the sample space is S={wm,bm,hm,am,om,wf,bf,hf,af,of}. The information given in the example can be summarized in the following table, called a two-way contingency table:

1. Since B={bm,bf}, P(B)=P(bm)+P(bf)=0.12+0.15=0.27.
2. Since MF={bf,hf,af,of}, P(M)=P(bf)+P(hf)+P(af)+P(of)=0.15+0.05+0.03+0.04=0.27
3. Since FN={wf,hf,af,of}, P(FN)=P(wf)+P(hf)+P(af)+P(of)=0.26+0.05+0.03+0.04=0.38

1. A box contains 10 white and 10 black marbles. Construct a sample space for the experiment of randomly drawing out, with replacement, two marbles in succession and noting the color each time. (To draw “with replacement” means that the first marble is put back before the second marble is drawn.)

2. A box contains 16 white and 16 black marbles. Construct a sample space for the experiment of randomly drawing out, with replacement, three marbles in succession and noting the color each time. (To draw “with replacement” means that each marble is put back before the next marble is drawn.)

3. A box contains 8 red, 8 yellow, and 8 green marbles. Construct a sample space for the experiment of randomly drawing out, with replacement, two marbles in succession and noting the color each time.

4. A box contains 6 red, 6 yellow, and 6 green marbles. Construct a sample space for the experiment of randomly drawing out, with replacement, three marbles in succession and noting the color each time.

5. In the situation of Exercise 1, list the outcomes that comprise each of the following events.

   1. At least one marble of each color is drawn.
   2. No white marble is drawn.

6. In the situation of Exercise 2, list the outcomes that comprise each of the following events.

   1. At least one marble of each color is drawn.
   2. No white marble is drawn.
   3. More black than white marbles are drawn.

7. In the situation of Exercise 3, list the outcomes that comprise each of the following events.

   1. No yellow marble is drawn.
   2. The two marbles drawn have the same color.
   3. At least one marble of each color is drawn.

8. In the situation of Exercise 4, list the outcomes that comprise each of the following events.

   1. No yellow marble is drawn.
   2. The three marbles drawn have the same color.
   3. At least one marble of each color is drawn.

9. Assuming that each outcome is equally likely, find the probability of each event in Exercise 5.

10. Assuming that each outcome is equally likely, find the probability of each event in Exercise 6.

11. Assuming that each outcome is equally likely, find the probability of each event in Exercise 7.

12. Assuming that each outcome is equally likely, find the probability of each event in Exercise 8.

13. A sample space is S={a,b,c,d,e}. Identify two events as U={a,b,d} and V={b,c,d}. Suppose P(a) and P(b) are each 0.2 and P(c) and P(d) are each 0.1.

    1. Determine what P(e) must be.
    2. Find P(U).
    3. Find P(V).

14. A sample space is S={u,v,w,x}. Identify two events as A={v,w} and B={u,w,x}. Suppose P(u)=0.22, P(w)=0.36, and P(x)=0.27.

    1. Determine what P(v) must be.
    2. Find P(A).
    3. Find P(B).

15. A sample space is S={m,n,q,r,s}. Identify two events as U={m,q,s} and V={n,q,r}. The probabilities of some of the outcomes are given by the following table:

    OutcomemnqrsProbablity0.180.160.240.21
    1. Determine what P(q) must be.
    2. Find P(U).
    3. Find P(V).

16. A sample space is S={d,e,f,g,h}. Identify two events as M={e,f,g,h} and N={d,g}. The probabilities of some of the outcomes are given by the following table:

    OutcomedefghProbablity0.220.130.270.19
    1. Determine what P(g) must be.
    2. Find P(M).
    3. Find P(N).

1. The sample space that describes all three-child families according to the genders of the children with respect to birth order was constructed in Note 3.9 "Example 4". Identify the outcomes that comprise each of the following events in the experiment of selecting a three-child family at random.

   1. At least one child is a girl.
   2. At most one child is a girl.
   3. All of the children are girls.
   4. Exactly two of the children are girls.
   5. The first born is a girl.

2. The sample space that describes three tosses of a coin is the same as the one constructed in Note 3.9 "Example 4" with “boy” replaced by “heads” and “girl” replaced by “tails.” Identify the outcomes that comprise each of the following events in the experiment of tossing a coin three times.

   1. The coin lands heads more often than tails.
   2. The coin lands heads the same number of times as it lands tails.
   3. The coin lands heads at least twice.
   4. The coin lands heads on the last toss.

3. Assuming that the outcomes are equally likely, find the probability of each event in Exercise 17.

4. Assuming that the outcomes are equally likely, find the probability of each event in Exercise 18.

1. The following two-way contingency table gives the breakdown of the population in a particular locale according to age and tobacco usage:

   A person is selected at random. Find the probability of each of the following events.

   1. The person is a smoker.
   2. The person is under 30.
   3. The person is a smoker who is under 30.

2. The following two-way contingency table gives the breakdown of the population in a particular locale according to party affiliation (A, B, C, or None) and opinion on a bond issue:

   A person is selected at random. Find the probability of each of the following events.

   1. The person is affiliated with party B.
   2. The person is affiliated with some party.
   3. The person is in favor of the bond issue.
   4. The person has no party affiliation and is undecided about the bond issue.

3. The following two-way contingency table gives the breakdown of the population of married or previously married women beyond child-bearing age in a particular locale according to age at first marriage and number of children:

   A woman is selected at random. Find the probability of each of the following events.

   1. The woman was in her twenties at her first marriage.
   2. The woman was 20 or older at her first marriage.
   3. The woman had no children.
   4. The woman was in her twenties at her first marriage and had at least three children.

4. The following two-way contingency table gives the breakdown of the population of adults in a particular locale according to highest level of education and whether or not the individual regularly takes dietary supplements:

   An adult is selected at random. Find the probability of each of the following events.

   1. The person has a high school diploma and takes dietary supplements regularly.
   2. The person has an undergraduate degree and takes dietary supplements regularly.
   3. The person takes dietary supplements regularly.
   4. The person does not take dietary supplements regularly.

Note: These data sets are missing, but the questions are provided here for reference.

1. Large Data Sets 4 and 4A record the results of 500 tosses of a coin. Find the relative frequency of each outcome 1, 2, 3, 4, 5, and 6. Does the coin appear to be “balanced” or “fair”?

2. Large Data Sets 6, 6A, and 6B record results of a random survey of 200 voters in each of two regions, in which they were asked to express whether they prefer Candidate A for a U.S. Senate seat or prefer some other candidate.

   1. Find the probability that a randomly selected voter among these 400 prefers Candidate A.
   2. Find the probability that a randomly selected voter among the 200 who live in Region 1 prefers Candidate A (separately recorded in Large Data Set 6A).
   3. Find the probability that a randomly selected voter among the 200 who live in Region 2 prefers Candidate A (separately recorded in Large Data Set 6B).

3. S={rr,ry,rg,yr,yy,yg,gr,gy,gg}

7. 1. {rr,rg,gr,gg}
   2. {rr,yy,gg}
   3. ∅

1. The relative frequencies for 1 through 6 are 0.16, 0.194, 0.162, 0.164, 0.154 and 0.166. It would appear that the die is not balanced.