Question

2. India and Australia play a series of 7 one-day matches. Each team has equal probability of
winning a match. No match ends in a draw. If the probability that India wins atleast three
consecutive matches can be expressed as P where p and q are relatively prime positive integers.
9
Find the unit digit of p.
The hunter
T
ton​

Answer 1

The probability that India will not win any game =
3
2
​
×
3
2
​
×
3
2
​
=
27
8
​

The probability that India will win atleast 1 game =1−the probability that India will not win any game =1−
27
8
​
=
27
19
​

Answer 2

Answer: The unit digit of p is 9.

Let's denote the probability of India winning a single match as x.

Then, the probability of India not winning a match can be represented as:                                    

                                     (1-x)

The probability of India winning exactly two consecutive matches can be represented as:

                                x^2 * (1-x)^5

The probability of India winning exactly three consecutive matches can be represented as:

                                x^3 * (1-x)^4

So, the probability of India winning at least three consecutive matches is:

                     x^3 * (1-x)^4 + x^4 * (1-x)^3 + x^5 * (1-x)^2 + x^6 * (1-x) + x^7

Since x and (1-x) are between 0 and 1, their unit digits will always be 0,1,5, or 6.

Hence, the unit digit of the expression above will be the same as the unit digit of x^7, which is 9.

So, the unit digit of p is 9.

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