Construct the probability distribution table for each of the following scenarios:
1.A Restaurant plays music from a playlist that contains 60% English songs and the rest are in Spanish, that are shuffled randomly.
When visiting the restaurant in 2 different occasions, X is the discrete random variable that represents the amount of Spanish songs played at the moment of entry
2. A fair six-sided die has the number 1 on one face, the number 2 on two faces and the number 3 on three faces.
X is the discrete random variable that stands for the sum of shown faces, when rolling the die twice
To construct the probability distribution table for each scenario, we need to calculate the probabilities of all possible outcomes of the random variables. Let's start with scenario 1:
1. Scenario: A Restaurant plays music from a playlist that contains 60% English songs and the rest are in Spanish, that are shuffled randomly. When visiting the restaurant in 2 different occasions, X is the discrete random variable that represents the amount of Spanish songs played at the moment of entry.
To construct the probability distribution table, we need to calculate the probabilities for all possible values of X.
Let's use the formula:
P(X = x) = nCx * p^x * (1-p)^(n-x)
where:
- n is the total number of songs played (2 in this case)
- x is the number of Spanish songs played
- nCx represents the number of combinations of x items chosen from a set of n items
- p is the probability of selecting a Spanish song (40% or 0.4 in this case)
- (1-p) is the probability of selecting an English song (60% or 0.6 in this case)
Let's calculate the probabilities for all possible values of X:
- When X = 0 (no Spanish songs played):
P(X = 0) = C(2, 0) * (0.4^0) * (0.6^(2-0))
= 1 * 1 * 0.36
= 0.36
- When X = 1 (one Spanish song played):
P(X = 1) = C(2, 1) * (0.4^1) * (0.6^(2-1))
= 2 * 0.4 * 0.6
= 0.48
- When X = 2 (two Spanish songs played):
P(X = 2) = C(2, 2) * (0.4^2) * (0.6^(2-2))
= 1 * 0.16 * 1
= 0.16
Now, we can construct the probability distribution table for scenario 1:
| X | 0 | 1 | 2 |
| P(X=x) | 0.36 | 0.48 | 0.16 |
2. Scenario: A fair six-sided die has the number 1 on one face, the number 2 on two faces, and the number 3 on three faces. X is the discrete random variable that stands for the sum of shown faces when rolling the die twice.
To construct the probability distribution table, we need to calculate the probabilities for all possible values of X.
The sum of the two dice will range from 2 to 6 (since the lowest sum is 1+1=2, and the highest sum is 3+3=6).
To calculate the probabilities, we can use a table to list all possible outcomes and their respective probabilities:
| Outcome | Sum | Probability |
| 1 | 2 | 1/36 |
| 2 | 3 | 2/36 |
| 3 | 4 | 3/36 |
| 4 | 5 | 4/36 |
| 5 | 6 | 5/36 |
| 6 | 7 | 6/36 |
| 7 | 8 | 5/36 |
| 8 | 9 | 4/36 |
| 9 | 10 | 3/36 |
| 10 | 11 | 2/36 |
| 11 | 12 | 1/36 |
Now, we have the probability distribution table for scenario 2:
| X | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| P(X=x) | 1/36 | 2/36 | 3/36 | 4/36 | 5/36 | 6/36 | 5/36 | 4/36 | 3/36 | 2/36 | 1/36 |
That's how you construct the probability distribution tables for the given scenarios.