Probability

[sathish]

Hi,

Please help me with this problem.

The probability of team A winning a test match against Team B is 1/3. Both the teams plan 6 test matches.

1. What is the probability of the Team A losing all test matches
2. What is the probability of Team A winning at least one test match.

Thanks

 

[pka]

The probability of team A winning a test match against Team B is 1/3. Both the teams plan 6 test matches.
1. What is the probability of the Team A losing all test matches
2. What is the probability of Team A winning at least one test match.

We must assume that these are independent binomial trials.
Clearly the probability that B wins is $\displaystyle 1-\tfrac{1}{3}=\tfrac{2}{3}$.

If a trial has probability $\displaystyle p$ of success and we have $\displaystyle N$ independent tries, then the probability of exactly $\displaystyle K,~0\le K\le N,$ successes is $\displaystyle \displaystyle\binom{N}{K}p^K(1-p)^{N-K}.$

Now you tell us what is next.

 

[sathish]

We must assume that these are independent binomial trials.
Clearly the probability that B wins is $\displaystyle 1-\tfrac{1}{3}=\tfrac{2}{3}$.

If a trial has probability $\displaystyle p$ of success and we have $\displaystyle N$ independent tries, then the probability of exactly $\displaystyle K,~0\le K\le N,$ successes is $\displaystyle \displaystyle\binom{N}{K}p^K(1-p)^{N-K}.$

Now you tell us what is next.

Hi pka,

Thanks for the formula. I will try applying it and will post here.

 

[soroban]

Hello, sathish!

The probability of team A winning a game against Team B is 1/3.
Both the teams plan 6 games.

$\displaystyle P(A\text{ wins}) \,=\,\dfrac{1}{3}\qquad P(A\text{ loses}) \,=\,\dfrac{2}{3}$

1. What is the probability of the Team A losing all 6 games?

$\displaystyle P(A\text{ loses 6 games}) \:=\:\left(\dfrac{2}{3}\right)^6 \:=\:\dfrac{64}{729}$

2. What is the probability of Team A winning at least one game?

$\displaystyle P(A\text{ wins at least one game}) \:=\:1 - \dfrac{64}{729} \:=\:\dfrac{665}{729}$