1.) a computer package sale comes with two different choices of printers and four different choices of monitors. if a store wants to display each package combination that is for sale, how many packages must be displayed? make a tree diagram showing the outcomes for selecting printer and monitor.
2.)your company routes all materials from NY to Chicago, then from Chicago to Denver, and then from Denver to L.A. There are four routes from NY to Chicago, six routes from Chicago to Denver, and three routes from Denver to LA. How many routes are there from NY to LA?
3.)you supervise 12 nurses and must assign one to head the ward on the first floor, one to head the ward on the second floor, one to head the ward on the third floor, and one to head the ward on the fourth floor. in how many ways can this be done?
4.)an urn contains 12 balls identical in every respect except color. there are 3 red balls, 7 green balls, and 2 blue balls. you draw two balls from the urn, but replace the first ball before drawing the second. find the probability that the first ball is red and the second is green. what is probability if repeating EXCEPT do not replace the first ball before drawing the second?
an urn contains 12 balls identical in every respect except color. there are 3 red balls, 7 green balls, and 2 blue balls. you draw two balls from the urn, but replace the first ball before drawing the second. find the probability that the first ball is red and the second is green. what is probability if repeating EXCEPT do not replace the first ball before drawing the second?
questions one
answer-5
1) To find the number of package combinations, we multiply the number of choices for each component. In this case, there are 2 choices of printers and 4 choices of monitors. Thus, the total number of package combinations that need to be displayed is 2 * 4 = 8.
Here is a tree diagram showing the outcomes for selecting printers and monitors for each package combination:
___P1___
| |
M1 4 4 M2
| |
P2 P2
/ | \ / | \
M1 4 4 M2 M1 4 4 M2
2) To find the number of routes from NY to LA, we multiply the number of routes for each leg of the journey. There are 4 routes from NY to Chicago, 6 routes from Chicago to Denver, and 3 routes from Denver to LA. Thus, the total number of routes from NY to LA is 4 * 6 * 3 = 72.
3) To assign one nurse to each floor, we multiply the number of choices for each floor. There are 12 nurses to choose from for the first floor, 11 nurses remaining for the second floor, 10 nurses remaining for the third floor, and 9 nurses remaining for the fourth floor.
Thus, the total number of ways to assign nurses to each floor is 12 * 11 * 10 * 9 = 11,880.
4) If we draw two balls with replacement, the probability of the first ball being red and the second being green is:
P(Red and Green) = P(Red) * P(Green) = (3/12) * (7/12) = 0.072.
If we draw two balls without replacement, after selecting a red ball, there are only 11 balls remaining in the urn. The probability of the second ball being green is:
P(Red and Green) = P(Red) * P(Green|Red) = (3/12) * (7/11) = 0.159.
1.) To determine the number of package combinations that must be displayed, multiply the number of choices for printers (2) by the number of choices for monitors (4). This can be represented using a tree diagram as follows:
- Printer 1
- Monitor 1
- Monitor 2
- Monitor 3
- Monitor 4
- Printer 2
- Monitor 1
- Monitor 2
- Monitor 3
- Monitor 4
So, there will be a total of 2 * 4 = 8 packages that need to be displayed.
2.) To calculate the total number of routes from NY to LA, multiply the number of routes from NY to Chicago (4), the number of routes from Chicago to Denver (6), and the number of routes from Denver to LA (3).
Number of routes from NY to LA = 4 * 6 * 3 = 72 routes.
3.) To determine the number of ways to assign a nurse to each floor, use the concept of permutations. Since each floor needs a different nurse, the permutation formula can be used:
Number of ways to assign a nurse to the first floor = 12 options
Number of ways to assign a nurse to the second floor = 11 options (one nurse has already been assigned to the first floor)
Number of ways to assign a nurse to the third floor = 10 options
Number of ways to assign a nurse to the fourth floor = 9 options
Using the formula for permutations, the total number of ways to assign nurses is:
12 * 11 * 10 * 9 = 11,880 ways.
4.) In this scenario, the first ball is replaced before drawing the second ball. Since the balls are replaced, the probability of drawing a certain ball remains the same for each draw.
The probability of drawing a red ball on the first draw is 3/12 (3 red balls out of 12 total balls).
The probability of drawing a green ball on the second draw is 7/12 (7 green balls out of 12 total balls).
Since the two events are independent, to find the combined probability, multiply the probabilities:
P(first ball is red and second is green) = (3/12) * (7/12) = 21/144.
If the first ball is not replaced before drawing the second ball, the total number of balls in the urn decreases by one for the second draw. Therefore, the probability calculations change accordingly.
The probability of drawing a red ball on the first draw is still 3/12 (3 red balls out of 12 total balls).
The probability of drawing a green ball on the second draw is now 7/11 (7 green balls out of remaining 11 balls).
So, the new combined probability is:
P(first ball is red and second is green without replacement) = (3/12) * (7/11) = 21/132.