Question

Question: Find the probability of winning a lottery by selecting the correct six integers, where the order in which these integers are selected does not matter, from the positive integers not exceeding.

(a) 30.

(b) 36.

(c) 42.

(d) 48.

Short answer

Answer

(a)The probability of winning a lottery by selecting the correct six integers, where the order in which these integers are selected does not matter, from the positive integers not exceeding 30 is$P\left(E\right)=\frac{1}{{}^{30}{C}_6}$.

(b)The probability of winning a lottery by selecting the correct six integers, where the order in which these integers are selected does not matter, from the positive integers not exceeding 36 is$P\left(E\right)=\frac{1}{{}^{36}C_6}$.

(c)The probability of winning a lottery by selecting the correct six integers, where the order in which these integers are selected does not matter, from the positive integers not exceeding 42 is$P\left(E\right)=\frac{1}{{}^{42}{C}_6}$.

(d) The probability of winning a lottery by selecting the correct six integers, where the order in which these integers are selected does not matter, from the positive integers not exceeding 48 is$P\left(E\right)=\frac{1}{{}^{48}{C}_6}$ .

Step-by-step solution

 Given  

Given are the six integers.

The Concept of probability

If$\mathit{S}$represents the sample space androle="math" $\mathit{E}$represents the event. Then the probability of occurrence of favourable event is given by the formula as below:

$\mathit{P}\mathbf{\left(}\mathit{E}\mathbf{\right)}\mathbf{=}\frac{\mathbf{n}\mathbf{\left(}\mathbf{E}\mathbf{\right)}}{\mathbf{n}\mathbf{\left(}\mathbf{S}\mathbf{\right)}}$.

The probability of winning a lottery (a)

As per the problem we have been asked to find the probability that a winning a lottery by selecting the correct six integers, where the order in which these integers are selected does not matter, from the positive integers not exceeding 30.

If$S$represents the sample space and$E$represents the event. Then the probability of occurrence of favourable event is given by the formula as below:$P\left(E\right)=\frac{n\left(I\right)}{n\left(S\right)}$

Sample space of numbers not exceeding 30 is$S=\{1,2,3,\dots ,30\}$

Total number of ways of selecting six integers out of 30 integers is${}^{30}{C}_6$.

There will be only one set of positive integers for which lottery can be drawn.

The probability of winning a lottery by selecting the correct six integers, where the order in which these integers are selected does not matter, from the positive integers not exceeding 30 is $P\left(E\right)=\frac{1}{{}^{30}{C}_6}$.

The probability of winning a lottery (b)

As per the problem we have been asked to find the probability that a winning a lottery by selecting the correct six integers, where the order in which these integers are selected does not matter, from the positive integers not exceeding 36.

If $S$represents the sample space and $E$represents the event. Then the probability of occurrence of favourable event is given by the formula as below:

$P\left(E\right)=\frac{m\left(K\right)}{n\left(S\right)}$

Sample space of numbers not exceeding 30 is$S=\{1,2,3,\dots ,36\}$

Total number of ways of selecting six integers out of 36 integers is${}^{36}{C}_6$.

There will be only one set of positive integers for which lottery can be drawn.

The probability of winning a lottery by selecting the correct six integers, where the order in which these integers are selected does not matter, from the positive integers not exceeding 36 is $P\left(E\right)=\frac{1}{{}^{36}{C}_6}$.

The probability of winning a lottery (c) 

As per the problem we have been asked to find the probability that a winning a lottery by selecting the correct six integers, where the order in which these integers are selected does not matter, from the positive integers not exceeding 42.

If $S$represents the sample space and $E$represents the event. Then the probability of occurrence of favourable event is given by the formula as below:

$P\left(E\right)=\frac{m\left(L\right)}{n\left(S\right)}$

Sample space of numbers not exceeding 42 is$S=\{1,2,3,\dots ,42\}$

Total number of ways of selecting six integers out of 42 integers isrole="math" localid="1668493722040" ${}^{42}{C}_6$.

There will be only one set of positive integers for which lottery can be drawn.

The probability of winning a lottery by selecting the correct six integers, where the order in which these integers are selected does not matter, from the positive integers not exceeding 42 is$P\left(E\right)=\frac{1}{{}^{42}{C}_6}$.

The probability of winning a lottery (d)

As per the problem we have been asked to find the probability that a winning a lottery by selecting the correct six integers, where the order in which these integers are selected does not matter, from the positive integers not exceeding 48.

If $S$represents the sample space and $E$represents the event. Then the probability of occurrence of favourable event is given by the formula as below:

$P\left(E\right)=\frac{m\left(L\right)}{n\left(S\right)}$

Sample space of numbers not exceeding 48 is$S=(1,2,3,\dots ,48)$

Total number of ways of selecting six integers out of 48 integers isrole="math" localid="1668493851821" ${}^{48}{C}_6$.

There will be only one set of positive integers for which lottery can be drawn.

The probability of winning a lottery by selecting the correct six integers, where the order in which these integers are selected does not matter, from the positive integers not exceeding 48 is$P\left(E\right)=\frac{1}{{}^{48}{C}_6}$.