Question

A football match may be either won, drawn or lost by the host country's team. So there are three ways of forecasting the result of any one match, one correct and two incorrect. Find the probability of forecasting at least three correct results for four matches.

Answer 1

We have given that the football match may be won, drawn or lost by the host team. Also there are two ways of forecasting the success of any one match, one correct and other two incorrect. So firstly we have to find the probability of forecasting correct and incorrect results. After that we have to ask for the probability of forecasting at least three correct results for four matches. At least three means, three and may be four.
So the probability will be the sum of probability of three correct and four correct.

Complete step by step solution:
Let ‘P’ be the probability that the result is correct.
So probability of correct result
$P = - \dfrac{1}{3}$
Let ‘q’ be the probability of an incorrect result.
So probability of incorrect result
$q = - \dfrac{2}{3}$
Now we have to find the probability of at least three correct results.
Since the number of matches are four. So there are a number of ways in which three correct results can take place.
${}^4{c_3}$
So probability of three correct results.
= {}^4{c_3}$$${\left( p \right)^3}{\left( q \right)^{4 - 3}}$$ Number ways in which four correct result can take place = {}^4{c_4}$So probability of three correct results. =${}^4{c_4}$$${\left( p \right)^4}{\left( q \right)^{4 - 4}}$=$\dfrac{{41}}{{31(4 - 3)}} \times {\left( {\dfrac{1}{3}} \right)^1}$\dfrac{1}{3} \times \dfrac{1}{3} \times \dfrac{1}{3} \times \dfrac{1}{3}$ =\dfrac{8}{{81}} + \dfrac{1}{{81}}$=$\dfrac{{8 + 1}}{{81}} = \dfrac{9}{{81}} = \dfrac{1}{9}$$

So probability of at least three correct result = $\dfrac{1}{9}$

Note:
Combination: - In combination, we arrange, but only in selecting is objects from n objects. We do not specify ordering in combination.
Probability: - It is a branch of mathematics concerning numerical descriptions of how likely an event is to occur or how likely it is that a proposition is true.