Finding a number that is smaller than 100 and that has more factors than 100.

capellitad9
2022-07-23
Answered

Finding a number that is smaller than 100 and that has more factors than 100.

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Let the sample space be

$S=1,2,3,4,5,6,7,8,9,10.$

Suppose the outcomes are equally likely. Compute the probability of the event E="an even number less than 9."

Suppose the outcomes are equally likely. Compute the probability of the event E="an even number less than 9."

asked 2022-06-26

Suppose we have 20 balls numbered 1,2,...,20. Each day we pick a single ball randomly.

1) What's the probability of picking all 20 balls in 22 days ?

2) What's the probability of picking only 1and2 in 4 days (Atleast 1 times each) ?

1) What's the probability of picking all 20 balls in 22 days ?

2) What's the probability of picking only 1and2 in 4 days (Atleast 1 times each) ?

asked 2022-06-12

Let there be a cube with $n$ sides denoted $1,...,n$ each. The cube is tossed $n+1$ times. For $1\le k\le n$ what is the probability that exactly $k$ first tosses give different number (i.e, the $(k+1)$-st toss give a number that was already gotten.) I really need to know why I got a slightly different answer from the official one.My attempt: Let us build a uniform sample space. $\mathrm{\Omega}=\{{a}_{i}=({i}_{1},...,{i}_{k})|1\le {i}_{j}\le n\}$. $|\mathrm{\Omega}|=(n+1{)}^{n}$, $\mathrm{\forall}\omega \in \mathrm{\Omega},P(\omega )=\frac{1}{|\mathrm{\Omega}|}$. We seek for the event $A=\{({i}_{1},...,{i}_{k},{i}_{k+1},...,{i}_{n+1})|{i}_{t}\ne {i}_{s},\mathrm{\forall}1\le t\ne s\le k,k\in \{{i}_{1},...,{i}_{k}\}\}$ This is the problematic part: $|A|={\textstyle (}\genfrac{}{}{0ex}{}{n}{k}{\textstyle )}\cdot k!\cdot k\cdot {n}^{n-k-1}$. (Then I and the answer use the formula for probability of an even it a uniform sample space.)The point is, the answer says: $|A|={\textstyle (}\genfrac{}{}{0ex}{}{n}{k}{\textstyle )}\cdot k!\cdot k\cdot {n}^{n-k}$ I don't understand why; First I pick $k$ numbers, count all their permutations, then pick one of them for the $(k+1)$-th toss, and then I have $n-k-1$ tosses left, each of which has $n$ possibilities.

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There are 5 women and 3 men waiting on standby for a flight to New York. Suppose
3 of these 8 people are selected at random, and a random variable X is defined to
be the number of women selected. Find Pr[X = 2].