The success rates of Cupid’s arrows

by Kurious Fox

To my dear master student……

Assume that there are 15 girls you are currently falling for and you lobbied Cupid to aim his arrows, attempting to make these girls reciprocate your feelings. Cupid can shoot the arrow only once to each girl for you. However, Cupid is getting a bit old and the success rate is p=0.7 instead of p = 1 like when he was young. ?

You want to know the number of girls that has the highest probability of success. We can compute these probabilities using the binomial probability formula which is:

P(X=r) = \binom{n}{r} * p^r * (1-p)^{n-r}

Where:

  • n is the number of trials (in this case, the number of girls you like, n=15)
  • r is the number of successes we are interested in (in this case, the number of girls that fall in love with you, r=0, 1, …, 15)
  • p is the probability of success on a single trial (in this case, the probability that Cupid’s arrow makes a girl fall in love with you, p=0.7)
  • C(n, r) is the combination of n things taken r at a time, and it is calculated as C(n, r) = n! / [r!(n-r)!]

Here are the probabilities:

  • P(X=0) = \binom{15}{0} \times (0.7^0) \times (0.3^{15})
  • P(X=1) = \binom{15}{1} \times (0.7^1) \times (0.3^{14})
  • …
  • P(X=15) = \binom{15}{15} \times (0.7^{15}) \times (0.3^0)

Here are the results:

XProbability
00.00000
10.00000
20.00003
30.00024
40.00131
50.00565
60.01992
70.05765
80.13659
90.26051
100.33327
110.27522
120.13982
130.04293
140.00757
150.00047

So, it looks like X = 10 has the highest probability of success (0.33). Ok, for X = 15, it falls down to 0.00047, but isn’t 10 is enough? ??

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