Question

In a hockey team there are 6 defenders, 4 offenders and 1 goalie, out of these, one player is to be selected randomly as a captain. Find the probability of the selection that –
1. The goalie will be selected
2. A defender will be selected

Answer 1

Hint: The first thing to do in these types of questions is to write the sample space and then remember the formula to calculate the probability of an event, the formula is,
$P(A)=\dfrac{n(A)}{n(S)}$
Now, P(A) is the probability of occurrence of Event A,
n(A) is the number of all outcomes in favour of event A,
n(S) is the number of all possible outcomes in the experiment or the number of elements in sample space.

Complete step-by-step answer:
From the question, we get that the sample space would include 6 defenders, 4 offenders and 1 goalie. So, we can denote them using letters D, O and G and specific subscripts. Thus we will have the sample space. Then, we can figure out the possible outcomes in each case and find the probability.

We know that sample space is the set of all possible outcomes of an experiment. The sample space for both the parts in the question is the same as we have to choose from the whole team and the team is the same in both the cases.
Let Dn be a random defender, On be a random offender and Gn be a random goalie.
Let us denote Sample space by the letter S.
S=\left\\{ {{D}_{1}},{{D}_{2}},{{D}_{3}},{{D}_{4}},{{D}_{5}},{{D}_{6}},{{O}_{1}},{{O}_{2}},{{O}_{3}},{{O}_{4}},{{G}_{1}} \right\\}
As every person is different (maybe on the basis of their performance) whether the position of playing is the same or not, so we give them numbers to distinguish from each other.
So the number of elements in the sample space, set S,
$n(S)=6+4+1=11$
1. Now in first part we want to find the probability of selecting a goalie as the captain of the team, as there is only one goalie (G1) so the number of favourable outcomes will be only one,
Let the event A be the event of selecting the goalie as the captain from the whole team.
So, number of elements in the set of favourable outcomes, set A,
A=\left\\{ {{G}_{1}} \right\\}
$n(A)=1$
Hence by putting the values of these terms in the original probability formula, we get the probability of occurrence of event A,
$P(A)=\dfrac{n(A)}{n(S)}=\dfrac{1}{11}$

2. In this case we want to find the probability of selecting a defender as the captain of the team, as there are six defenders out of which we have to select one but the choices available to us are six,
Let the event A be the event of selecting a defender as the captain from the whole team.
So, number of elements in the set of favourable outcomes, set A,
A=\left\\{ {{D}_{1}},{{D}_{2}},{{D}_{3}},{{D}_{4}},{{D}_{5}},{{D}_{6}} \right\\}
$n(A)=6$
Hence by putting the values of these terms in the original probability formula, we get the probability of occurrence of event A,
$P(A)=\dfrac{n(A)}{n(S)}=\dfrac{6}{11}$

Note: Remember probability is a possibility and when we are calculating it we look for possible favourable choices to put in the numerator rather than just putting the choice we want as in the second case we want one captain but we have 6 choices, we write 6 in numerator not 1 as we have 6 choices who are eligible of becoming a captain and to choose from.
Do not forget to write the sample space first as it will lower your chances of mistakes and make the question easy for you to solve.
We can use permutation and combination instead of just writing numbers for difficult problems or the lengthy ones.