How frequently do streaks of luck come along?
If I repeatedly flip a coin ๐ช, any fule kno it takes on average two tosses to get a head. Or six rolls of a dice ๐ฒ on average to get a six.
What about two heads in a row? Or, two sixes in a row, or three or four...?
A #random๐งต
If I repeatedly flip a coin ๐ช, any fule kno it takes on average two tosses to get a head. Or six rolls of a dice ๐ฒ on average to get a six.
What about two heads in a row? Or, two sixes in a row, or three or four...?
A #random๐งต
At first guess you might think it takes 4 tosses on average to get two heads in a row, since the probability of getting two heads on any two consecutive tosses is 1 in 4.
But this is too small. Being twice the time for a single head, it is the expected time to get two heads...
But this is too small. Being twice the time for a single head, it is the expected time to get two heads...
...๐ฏ๐ฐ๐ต ๐ฏ๐ฆ๐ค๐ฆ๐ด๐ด๐ข๐ณ๐ช๐ญ๐บ ๐ค๐ฐ๐ฏ๐ด๐ฆ๐ค๐ถ๐ต๐ช๐ท๐ฆ.
So 4 is too small.
You might also guess at 8, since if we were to divide the sequence into pairs of coin tosses, each has 1/4 of being two heads, so it should take on average 4 pairs of tosses.
๐๐ ๐๐ ๐๐ ๐๐
So 4 is too small.
You might also guess at 8, since if we were to divide the sequence into pairs of coin tosses, each has 1/4 of being two heads, so it should take on average 4 pairs of tosses.
๐๐ ๐๐ ๐๐ ๐๐
But this ignores the fact that the first two consecutive heads might not align with the way we grouped them into pairs:
๐๐ ๐๐ ๐๐ ๐๐ ๐๐
So 8 is too big. The true answer has to be between 4 and 8.
In fact, it can be shown that it takes, on average, ๐ฒ tosses.
๐๐ ๐๐ ๐๐ ๐๐ ๐๐
So 8 is too big. The true answer has to be between 4 and 8.
In fact, it can be shown that it takes, on average, ๐ฒ tosses.
Similarly it takes ๐ฐ๐ฎ dice rolls on average to get two consecutive sixes!
This is more than the naรฏve guess of 6ยฒ=36.
Going further...it takes 14 tosses on average to get three consecutive heads of a coin, and 258 rolls of a dice for three consecutive sixes.
Spot the pattern?
This is more than the naรฏve guess of 6ยฒ=36.
Going further...it takes 14 tosses on average to get three consecutive heads of a coin, and 258 rolls of a dice for three consecutive sixes.
Spot the pattern?
The expected time to get 1,2 and 3 consecutive heads are respectively:
2, 2ยฒ+2 = 6, 2ยณ+2ยฒ+2 = 14
Expected time to get 1,2 and 3 consecutive sixes of a dice are respectively:
6, 6ยฒ+6 = 42, 6ยณ+6ยฒ+6 = 258
Can you guess how long it takes to get six sixes on average?
2, 2ยฒ+2 = 6, 2ยณ+2ยฒ+2 = 14
Expected time to get 1,2 and 3 consecutive sixes of a dice are respectively:
6, 6ยฒ+6 = 42, 6ยณ+6ยฒ+6 = 258
Can you guess how long it takes to get six sixes on average?
Did you get it?
6โถ+6โต+6โด+6ยณ+6ยฒ+6 = 55,986
Well done!
6โถ+6โต+6โด+6ยณ+6ยฒ+6 = 55,986
Well done!
The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will eventually type the complete works of William Shakespeare.
en.wikipedia.org
en.wikipedia.org
Being a little less ambitious, how long before it types the word ABRACADABRA?
We suppose it hits each letter A through Z independently with equal probability, and ignore any other characters.
We suppose it hits each letter A through Z independently with equal probability, and ignore any other characters.
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