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 find the probability distribution of the random variable X (the number of Spanish songs played at the moment of entry), we need to consider the probability of each possible value of X.
Since the playlist contains 60% English songs and the rest are Spanish, we can determine the probability of getting a certain number of Spanish songs using the binomial distribution.
The binomial distribution is used in situations where there are only two possible outcomes (Spanish song or English song) and each outcome has a fixed probability (40% for Spanish song and 60% for English song).
To calculate the probability distribution, we need to know the number of trials (number of songs played at the moment of entry) and the probability of success (getting a Spanish song). Let's assume that on each occasion, 30 songs were played at the moment of entry.
Using the binomial distribution formula, the probability of getting exactly x Spanish songs out of 30 is given by:
P(X = x) = (30C x) * (0.4^x) * (0.6^(30-x))
Where "30C x" represents the combination formula for selecting x items from a set of 30 (calculated as 30! / (x! * (30-x)!)), and "0.4" and "0.6" represent the probabilities of success (getting a Spanish song) and failure (getting an English song), respectively.
To find the probability distribution for all possible values of X (ranging from 0 to 30), we can calculate P(X = x) for each value of x.
Here is a table showing the probability distribution for the number of Spanish songs (X) when 30 songs are played at the moment of entry:
X | P(X = x)
--------------
0 | 0.000001
1 | 0.000055
2 | 0.000830
3 | 0.006248
4 | 0.031241
5 | 0.109368
6 | 0.273421
7 | 0.495669
8 | 0.679899
9 | 0.801235
10 | 0.871211
11 | 0.921302
12 | 0.953059
13 | 0.973760
14 | 0.986299
15 | 0.993576
16 | 0.997028
17 | 0.998910
18 | 0.999647
19 | 0.999911
20 | 0.999983
21 | 0.999997
22 | 0.999999
23 | 1.000000
24 | 0.999999
25 | 0.999997
26 | 0.999990
27 | 0.999976
28 | 0.999947
29 | 0.999889
30 | 0.999778
Please note that the probabilities have been rounded to six decimal places for simplicity.