Please show all steps to the solution. Thank you! Take Home Problem 2 (18 points

Question

Please show all steps to the solution. Thank you!

Take Home Problem 2 (18 points) By a process to be described later, we will randomly form a 7-digit number where Only the last digit (ones place) can be zero all 7 digits must be different. Consider the sample space S consisting of all possible 7-digit numbers satisfying the above constraints All 7-digit numbers sati sfying: s- all digits different only the last digit can be zero The sample space above is partitioned (divided or separated) into the 7-digit numbers that end in zero and those 7-digit numbers that do not end in zero All 7-digit numbers satisfying: l all digits different last digit is zero All 7-digit numbers satisfying all digits different no digits are zero Since the sample space contains literally thousands of elements, it is unreasonable to list them all. It shall be your task to count them:

Explanation / Answer

(A) Total number of elements in sample space

as there can be 9 digits which can come at the first place, and 8 number on second and so on but at last digit we can have four number as 0 can also be included.

Total number of elements in sample space = 9 * 8 * 7 * 6 * 5 * 4 * 4 = 241920

Number that contians 0 in sample space = 9 * 8 * 7 * 6 * 5 * 4 * 1 = 60480

Number that not contains 0 in sample space = 9 * 8 * 7 * 6 * 5 * 4 * 3 = 181440

Number that are odd here , as any of the 10 digits can come at last place so there is equal numbers of odd and even number in sample space

so odd numbers are = 120960

Even numbers are = 120960

(b) Here we have to seperatly draw the probability of number who contains zero in the end and those who don't.

as

Pr(a number will contain zero) = Pr(a number doesn't contain zero) = 1/2

so as there are 1814440 such number which doesn't contain zero. So,

Pr(to get an specific number which doesn't contain zero) = 1/(2 * 181440) = 1/362880

Similary, there are 60480 such number which contains zero. so,

Pr(to get an specific number which contains zero) = 1/(2 * 60480) = 1/120960

(C)

Here we have to find the probability that a number be odd

Pr(odd) = Pr(Ends with 1,3,5,7 or9) = Pr(To get a specific number which doesn't end in zero) * Total odd numbers

= 1/362880 * 120960 = 1/3

Pr(The 7 digit number will be multiple of 5) = Pr(Ends with 0) * Pr(Total such numbers) + Pr(Ends with 5) * total such numbers

= 1/120960 * 60480 + 1/(362880) * (120960/5) = 1/2 + 1/15 = 17/30

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