Question

You are at a European beach with 60 other visitors. 36 of them speak English. If you randomly meet two people on the beach, what is the probability that they both speak English

Answer 1

Answer:

Assuming I'm one of the 36 English speakers and the other 24 speak Spanish for illustration purposes.  The problem can be modeled as 59 marbles with 35 E and 24 S marbles as N = 59 possible outcomes = n(E) + n(S) = 35 + 24.

So I reach into the pile of marbles (on the beach) and the probability that it's p(E) = n(E)/N = 35/59 = 0.593220339 when I meet the one person. ANS

I assume I remember that first person; so I remove him from the marbles (by avoiding him on the beach) and now my probability is p(E and E) = n(E)/N * n(E)-1/(N - 1) = 35/59*34/58 = 0.347749854 ANS

Following the same logic p(E and E and E) = 35/59*34/58*33/57 = 0.201328863 ANS

This last one is different from the first three.  This one is p(E >= 1|4 attempts).  We can trace out a probability tree to identify those branches that contain at least one E event.  So:

EEEE p() = 35/59 * 34/58 * 33/57 * 32/56 =  

EEES p() = 35/59 * 34/58 * 33/57 * 24/56 =

EESE p() = 35/59 * 34/58 * 24/57 * 33/56 =

ESEE p() = 35/59 * 24/58 * 34/57 * 33/56 =

SEEE p() = 24/59 * 35/58 * 34/57 * 33/56 =

EESS p() = 35/59 * 34/58 * 24/57 * 23/56 =

ESES p() = 35/59 * 24/58 * 34/57 * 23/56 =

SEES

SESE

SSEE

ESSS  And so on, but...a big BUT...why do all this when

SESS

SSES

SSSE

SSSS

p(E>=1|4) = 1 - p(S and S and S and S) = 1 - 24/59 * 23/58 * 22/57 * 21/56 = 0.976652619   ANS.  In other words we find the probability of not meeting an Englishman and take 1 minus that value to find the probability of meeting at least one.

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Step-by-step explanation: