WHY ALL THE BUSES GO FOR NORTH DIRECTION, WHEN YOU WANT TO GO TO SOUTH?

We wait at a bus stop. We want to go for left direction. But all the buses seem to go towards right. Why? This paradox can be explained using statistics.

What is the chance of getting a head in a coin toss? It is 1/2 or .5. What is the chance of getting 3 heads one after another? It is 0.5*0.5*0.5 or 0.5^3. Hence the chance of getting k heads successively is 0.5^k. If I toss the coin 100 times, how many times 'consecutive 3 heads" will occur.

It is 0.5^3*100 =12.5 At least 12 times '3 heads' will occur in a row.

Finally, the chance of getting string of k heads in n tosses is

e = 0.5^k*n

as the K increases, e decreases.

for example, if k=6, n=100

e = 0.5^6*100 = 1.56

'Row of 6 heads' will occur only one time in 100 tosses. I want to find out what is the maximum value of k (number of heads) that will occur at least once (e=1) in 'n' tosses.

again

e= 0.5^k *n

1=0.5^k*n

using math

K(max) = log (n)/log (2).

we got the formula

Normally, equal number of buses go back and forth. Hence the chance of getting a bus for your destination or the other bus is 0.5.

Let us assume that 300 buses go on your road in a time window. Maximum how many buses will go in opposite direction one after another? Let us use the formula

k(max) = Log 300/log 2

K= 8.22

At least 8 buses and at least once will go in the opposite direction frustrating you. By bad luck, you selected that unfortunate time. If you wait little longer, your bus will arrive. So, patience pays.

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