Question

Whenever horses \[a,\,b,\,c\] race together, their respective probabilities of winning the race are \[0.3,\,\,0.5\] and \[0.2\] respectively. If they race three times the probability that “the same horse wins all the three races” and the probability that \[a,\,b,\,c\] each wins one race, are respectively (Assume no dead heat)
A. \[\dfrac{8}{{50}},\,\dfrac{9}{{50}}\]
B. \[\dfrac{{16}}{{100}},\,\dfrac{3}{{100}}\]
C. \[\dfrac{{12}}{{50}},\,\dfrac{{15}}{{50}}\]
D. \[\dfrac{{10}}{{50}},\,\dfrac{8}{{50}}\]

Answer 1

Hint:The question has two parts. Solve the two parts separately. For the first part we need to find the probability that the same horse wins all the races, consider all the possible cases for it and find the total probability. In the second part we need to find the probability that all horses win one race each, for this first find the number of cases, find the probability for each case then find the required probability.

Complete step by step solution:
Given, the probability of horse \[a\] to win the race is \[P(a) = 0.3\]
The probability of horse \[b\] to win the race is \[P(b) = 0.5\]
The probability of horse \[c\] to win the race is \[P(c) = 0.2\]

As, the question has two parts, we will solve each part one by one
In the first part, we are asked to find the probability that the same horse wins all the three races when the horses race three times.

There can be three cases
(1) Horse \[a\] wins all the races (2) Horse \[b\] wins all the races (3) horse \[c\] wins all the races

For case (1) when horse \[a\] wins all the three races. The probability will be,
\[P(a\_3) = P(a).P(a).P(a)\]

Putting the value of \[P(a)\] we get,
\[P(a\_3) = 0.3 \times 0.3 \times 0.3\]
\[ \Rightarrow P(a\_3) = 0.027\] (i)

For case (2) when horse \[b\] wins all the three races. The probability will be,
\[P(b\_3) = P(b).P(b).P(b)\]

Putting the value of \[P(b)\] we get,
\[P(b\_3) = 0.5 \times 0.5 \times 0.5\]
\[ \Rightarrow P(b\_3) = 0.125\] (ii)

For case (3) when horse \[c\] wins all the three races. The probability will be,
\[P(c\_3) = P(c).P(c).P(c)\]

Putting the value of \[P(c)\] we get,
\[P(c\_3) = 0.2 \times 0.2 \times 0.2\]
\[ \Rightarrow P(c\_3) = 0.008\] (iii)

The cases (1), (2) and (3) are mutually exclusive events that mean they cannot occur together or at the same time and for such events total probability is addition of all the events. So, here total probability will be,
\[P = P(a\_3) + P(b\_3) + P(c\_3)\] (iv)

Using equations (i), (ii) and (iii) in (iv) we get,
\[P = 0.027 + 0.125 + 0.008\]
\[ \Rightarrow P = \dfrac{{27}}{{1000}} + \dfrac{{125}}{{1000}} + \dfrac{8}{{1000}}\]
\[ \Rightarrow P = \dfrac{{160}}{{1000}}\]
\[ \Rightarrow P = \dfrac{8}{{50}}\]

In the second part, we are asked to find the probability that each horse \[a,\,b,\,c\] wins one race when they race three times.

Total number of cases each horse can win a race will be the number of ways we can arrange the three horses which is \[3!\]. Therefore, the total number of cases will be,
\[C = 3! = 3 \times 2 = 6\]

Each case will have the probability,
\[P({\text{each}}\,{\text{case}}) = P(a).P(b).P(c)\]

Putting the value \[P(a)\], \[P(b)\] and \[P(c)\], we get
\[P({\text{each}}\,{\text{case}}) = 0.3 \times 0.5 \times 0.2\]

This is the probability for one case and the number of cases is \[6\]. So, for \[6\] cases the probability will be or the total probability will be,

\[P' = 6 \times P({\text{each}}\,{\text{case}})\]

Putting the value of \[P({\text{each}}\,{\text{case}})\], we get
\[P' = 6 \times 0.3 \times 0.5 \times 0.2\]
\[ \Rightarrow P' = 6 \times \dfrac{3}{{10}} \times \dfrac{5}{{10}} \times \dfrac{2}{{10}}\]
\[ \Rightarrow P' = 6 \times \dfrac{{30}}{{1000}}\]
\[ \Rightarrow P' = \dfrac{9}{{50}}\]

Therefore, for the first part the probability is \[\dfrac{8}{{50}}\] and for the second part the probability is \[\dfrac{9}{{50}}\].

Hence, the correct answer is option (A) \[\dfrac{8}{{50}},\,\dfrac{9}{{50}}\]

Note:Two important terms you should remember in case of probability are mutually exclusive events and mutually inclusive events. Mutually exclusive events do not occur at same time, for example when a coin is tossed the result can be head or tail, both cannot occur together whereas mutually inclusive events cannot occur independently.