All of us, at least once in our life, have been in a particular social activity called dancing party. It means that, given a promising good evening, you made a decision of having fun in a certain place where people decided to follow the rhythm of the music in the best possible way. This decision came from multiple reasons: cause you have some friends who told you to go and have fun, or cause you wanted to go to meet your friends, or you are new in town and want to explore new places, or you are taking dancing lessons and want to practice somewhere all the things you just learnt.
Once you are up for some nice dancing time, which means that you have selected the proper place with a good dance floor, there are some other crucial factors to take into account in order to have a good time. Each of these factors intrinsically has a certain probability to happen, let list them in detail and in order of occurrence:
- The rhythm of the song: The DJ (or band) typically plays one or few types of (somehow) similar rhythms in a party. For example: (i) tango, tango vals and milonga; (ii) salsa, bachata and kizomba; etc.
In a typical situation, let assume that there is a DJ playing songs randomly from a playlist, and the list has M songs. Assume you want to dance with a stranger, which is the most critical situation (since you don't know the partner's preferences). Assume also that the partner always has 50% chance (Yes or No) to accept dancing with you (related to different factors such as the impression you give to the partner, how tired the partner is, how late the evening is,...) and this chance is present in every song. And assume that you know how to dance all the rhythms. You browse for a possible partner to dance with, and I guess you want to calculate the probability to dance, denoted P, the first song you ask for. Then you might have different scenarios:
Scenario 1A: The partner knows how to dance all the rhythms.
The P value is simply the 50% chance to get accepted by the partner: P=0.5.
Scenario 2A: The partner only knows how to dance n songs from the DJ list (where n <= M).
In this case P can be calculated using the specific multiplication rule, which is valid in this case since the two events (type of song the partner is able to dance, and the partner answer) are independent, therefore:
P = (n/M)*(1/2)= n/(2M)
Note that P in scenario 2 is lower or equal to P in scenario 1. Is equal when n=M.
Scenario 1B: You succeed to enjoy scenario 1A and want to dance the next song with the partner.
The P value is calculated by the multiplication of chances given by the decision of the partner. Then:
P = 0.5*0.5 = 0.25.
Note that for more songs, the P value will decrease towards zero (since the partner is getting tired, is getting late,...
Scenario 2B: You succeed to enjoy scenario 1B and want to dance the next song with the partner.
We still can use the multiplication rule probability, but in this case we have:
P(a ∩ b) = P(a) P(b|a), where P(a) is the probability that event a happens, and P(b|a) is the probability that event b happens given a. In our case, P(a) has already been calculated in scenario 2A (i.e. n/2M), now (assuming that a song is played only once), the second probability (without including the final decision of the partner, which is 50%) is then (n-1)/(M-1), all together gives:
P = (n/M)*(1/2)*((n-1)/(M-1))*(1/2) = (n*(n-1))/(M*(M-1)*4)
Note that P is decreasing faster towards zero than in scenario 2A, meaning that in this situation you will rapidly have no chance to dance a certain song with your current partner.
More about the multiplication rule probability can be found here, and about conditional probability here.
After these numbers, in the middle of the party, I guess you still has the courage to go and invite somebody to dance. Don't think twice and go!
See you around!
Jesús
- The rhythm of the song: The DJ (or band) typically plays one or few types of (somehow) similar rhythms in a party. For example: (i) tango, tango vals and milonga; (ii) salsa, bachata and kizomba; etc.
In a typical situation, let assume that there is a DJ playing songs randomly from a playlist, and the list has M songs. Assume you want to dance with a stranger, which is the most critical situation (since you don't know the partner's preferences). Assume also that the partner always has 50% chance (Yes or No) to accept dancing with you (related to different factors such as the impression you give to the partner, how tired the partner is, how late the evening is,...) and this chance is present in every song. And assume that you know how to dance all the rhythms. You browse for a possible partner to dance with, and I guess you want to calculate the probability to dance, denoted P, the first song you ask for. Then you might have different scenarios:
Scenario 1A: The partner knows how to dance all the rhythms.
The P value is simply the 50% chance to get accepted by the partner: P=0.5.
Scenario 2A: The partner only knows how to dance n songs from the DJ list (where n <= M).
In this case P can be calculated using the specific multiplication rule, which is valid in this case since the two events (type of song the partner is able to dance, and the partner answer) are independent, therefore:
P = (n/M)*(1/2)= n/(2M)
Note that P in scenario 2 is lower or equal to P in scenario 1. Is equal when n=M.
Scenario 1B: You succeed to enjoy scenario 1A and want to dance the next song with the partner.
The P value is calculated by the multiplication of chances given by the decision of the partner. Then:
P = 0.5*0.5 = 0.25.
Note that for more songs, the P value will decrease towards zero (since the partner is getting tired, is getting late,...
Scenario 2B: You succeed to enjoy scenario 1B and want to dance the next song with the partner.
We still can use the multiplication rule probability, but in this case we have:
P(a ∩ b) = P(a) P(b|a), where P(a) is the probability that event a happens, and P(b|a) is the probability that event b happens given a. In our case, P(a) has already been calculated in scenario 2A (i.e. n/2M), now (assuming that a song is played only once), the second probability (without including the final decision of the partner, which is 50%) is then (n-1)/(M-1), all together gives:
P = (n/M)*(1/2)*((n-1)/(M-1))*(1/2) = (n*(n-1))/(M*(M-1)*4)
Note that P is decreasing faster towards zero than in scenario 2A, meaning that in this situation you will rapidly have no chance to dance a certain song with your current partner.
More about the multiplication rule probability can be found here, and about conditional probability here.
After these numbers, in the middle of the party, I guess you still has the courage to go and invite somebody to dance. Don't think twice and go!
See you around!
Jesús