Three people are to be selected at random, each will be given one gift card.
There is one card from Home Depot,one from Best Buy and one from Red Lobster. The first person selected will get to choose between the two remaining cards. The third person selected gets the third card.
a) Determine the number of points in the sample space.
b) construct a tree diagram and determine the sample space.
Determine the probability that
a) the best buy card is selected first
b)the home depot card is selected first and the red lobster card is selected last
c)the cards are selected in this order best buy, red lobster, home depo
a) The number of points in the sample space can be determined using the formula for counting the outcomes of multiple events. In this case, there are 3 people who can be selected first, then 2 remaining people who can be selected second, and finally 1 person remaining who is selected last. So the number of points in the sample space is: 3 x 2 x 1 = 6.
b) Here is a tree diagram representing the sample space:
___________1st person (3 choices)_________
/ \
Home Depot Best Buy
| |
| |
___________2nd person (2 choices)_________
/ \
Home Depot Best Buy
/ |
| |
___________3rd person (1 choice)__________
/ \
Home Depot Red Lobster
| |
| |
___________End of Tree Diagram_________
The sample space includes all possible combinations of people receiving gift cards: Home Depot, Best Buy, Red Lobster, Home Depot, Best Buy, Red Lobster.
c) The probability of the cards being selected in the order Best Buy, Red Lobster, Home Depot is calculated by dividing the number of favorable outcomes by the total number of outcomes in the sample space. In this case, there is only one favorable outcome: Best Buy, Red Lobster, Home Depot. So the probability is 1/6.
a) To determine the number of points in the sample space, we need to consider the number of all possible outcomes. In this scenario, there are 3 people who can be selected first, 2 people for the second selection, and 1 person for the third selection. Therefore, the total number of possible outcomes is 3 × 2 × 1 = 6.
b) The tree diagram for the sample space can be constructed as follows:
Person 1 (3 options)
/ | \
Home Depot (2) Best Buy (2) Red Lobster (2)
| | |
Person 2 (2) Person 2 (1) Person 2 (1)
/ \
Home Depot (1) Red Lobster (1)
The sample space, based on the tree diagram, is:
(Home Depot, Best Buy, Red Lobster)
(Home Depot, Red Lobster, Best Buy)
(Best Buy, Home Depot, Red Lobster)
(Best Buy, Red Lobster, Home Depot)
(Red Lobster, Home Depot, Best Buy)
(Red Lobster, Best Buy, Home Depot)
c) To determine the probability of certain events, we need to count the number of favorable outcomes and divide it by the total number of possible outcomes.
a) The probability of the Best Buy card being selected first is 2 out of 6, as there are two outcomes in the sample space where Best Buy is selected first: (Best Buy, Home Depot, Red Lobster) and (Best Buy, Red Lobster, Home Depot). Therefore, the probability is 2/6 = 1/3.
b) The probability of the Home Depot card being selected first and the Red Lobster card being selected last is 1 out of 6. There is only one outcome in the sample space where this condition is met: (Home Depot, Best Buy, Red Lobster). Therefore, the probability is 1/6.
c) The probability of the cards being selected in the order Best Buy, Red Lobster, Home Depot is also 1 out of 6. There is only one outcome in the sample space where this order is met: (Best Buy, Red Lobster, Home Depot). Therefore, the probability is 1/6.
To answer the questions, let's break down the problem step by step:
a) Determine the number of points in the sample space.
In this scenario, we need to find the total number of possible outcomes when selecting three people. Since each person can only receive one gift card, the total number of outcomes can be calculated using the concept of permutations, specifically the permutation formula for selecting r objects from n objects without replacement:
nPr = n! / (n-r)!
In our case, n = 3 (because there are 3 people) and r = 3 (because each person needs to be selected). Plugging these values into the formula, we get:
3P3 = 3! / (3-3)! = 3! / 0! = 3! = 3 x 2 x 1 = 6
Therefore, there are 6 different possible outcomes in the sample space.
b) Construct a tree diagram and determine the sample space.
A tree diagram is an effective way to visually represent the different possible outcomes in this scenario. Start by drawing a tree trunk with three branches, representing the three people to be selected. Then, for each branch, draw two sub-branches representing the two choices for the gift card.
The resulting tree diagram would look like this:
(P1)
/ \
(BB) (HD)
| |
(P2) (P2)
/ \ / \
(RL) (HD) (RL) (BB)
| | |
(P3) (P3) (P3) (P3)
The sample space can be determined by listing all the possible combinations for the three people receiving the gift cards:
- (P1, BB, RL)
- (P1, BB, HD)
- (P1, HD, RL)
- (P1, HD, BB)
- (P1, RL, HD)
- (P1, RL, BB)
Therefore, the sample space consists of the above 6 possible outcomes.
Now, let's move on to the probabilities:
a) Probability of the Best Buy card being selected first.
In the sample space, there are three outcomes where the Best Buy card is selected first: (P1, BB, RL), (P1, BB, HD), (P1, BB, RL). So, the probability is 3/6, which simplifies to 1/2.
b) Probability of the Home Depot card being selected first and the Red Lobster card being selected last.
In the sample space, there is only one outcome where the Home Depot card is selected first and the Red Lobster card is selected last: (P1, HD, RL). So, the probability is 1/6.
c) Probability of the cards being selected in the order: Best Buy, Red Lobster, Home Depot.
In the sample space, there is no outcome where the cards are selected in this specific order. Therefore, the probability is 0/6, which simplifies to 0.
Please note that the probabilities mentioned above assume that the selection process is completely random and each person is equally likely to be chosen at each step.