High school probability Uncovered: Discover the Secrets with Plainmath's Expert Help

In a horse race, how many different finishes among the first three places are possible for a ten-horse race?

How do you evaluate 8P3?

A six sided die is rolled six times. What is the probability that each side appears exactly once?

In a raffle 5 winners get to chose from 5 prizes starting with the fist name drawn. iIf 87 people entered the raffle how many ways can the winners be arranged?

50 students traveled to Europe, last year. Of these, 12 visited Amsterdam, 13 went to Berlin, and 15 were in Copenhagen. Some visited two cities: 3 visited both Amsterdam and Berlin, 6 visited Amsterdam and Copenhagen, and 5 visited Berlin and Copenhagen. But only 2 visited all three cities.
Question : How many students visited Copenhagen, but neither Amsterdam nor Berlin?
I have done the graph and I think the answer is 6 but I would like to learn how to compute it.

I am trying to improve my stat skills from a book that gives an exercise, I cannot get my head around.<br.It goes like this:<br.There are 5 green balls and 1 red ball in a box. We draw four times randomly with replacement. What is the probability that we draw at least two red balls?<br.My guess would be<br.(1/6)^2 + (1/6)^3 + (1/6)^4 because the probability of at least 2 red balls out of 4 draws must be equal to drawing two or three or four times a red ball. Hence I would use the addition rule, which gives 0.03 = 3 percent<br.However, the book says it is 13 percent, and gives the following explanation (150+20+1)/1296.<br.Why is this so?<br.My attempt to trace back the terms:<br.I can possibly see how it got the first and third term in the fraction: The first could be (5/6)^2 = 25/36, which is the probability of drawing two green balls; and the third could be (1/6)^4 = 1/1296, which is the probability of drawing four red balls. I have no idea though where they could possibly get the 20 from.

How many ways can 14 books be organized on a shelf?

We have a condensed deck of 40 cards containing only the denominations from Ace through 10. We draw four cards uniformly at random (without replacement).
1. What is the probability that we draw four distinct denominations, such as one Ace, one five, one six and one seven.
2. What is the probability that we draw at least three hearts.
For 1, let's say we divide the the 40 cards into 10 piles of 4 cards each (of each kind Ace to 10). Then there are 10C4 ways to choose 4 piles and 4C1 ways to choose a card from each pile, so the probability should be ((10C4)(4)(4C1))/(40C4) = 0.036. Is this correct or am I missing something?
For 2, I think something like 1 - Pr(choosing exactly 2 hearts) could work, but I don't know where to go from here.

For the experiment of tossing a six sided die twice, what is the probability that the sum is 2, 3 or 12?

Given the probability $p=87/100$ that an event will not happen, how do you find the probability that the event will happen?

How do you find the probability of selecting a face card from a standard deck of 52 playing cards?

What is the probability that Mary will get on the bus first if Jerry, Kerry, Larry, and Mary are waiting at a bus stop?

How many four letter words are possible using the first 5 letters of the alphabet if letters can not be repeated?

I know the independent concept and multiplication rules.

Example Flip Coin

5 times flip, how many times success of Head?
Ans : $\frac{1}{2}\cdot <!--\; \cdot \; -->\frac{1}{2}\cdot <!--\; \cdot \; -->\frac{1}{2}\cdot <!--\; \cdot \; -->\frac{1}{2}\cdot <!--\; \cdot \; -->\frac{1}{2}=\frac{1}{32}$

Why do we use multiplication rules, not other rules?
I know if we use addition then probability 2.5 which is invalid because probability should be between 0 to 1. Same for Subtraction and division.
But, I am still confused why we use multiplication? What is the basic and core logic behind this?
Can anyone tell me the concept of this?

In how many ways can 5 letters be mailed if there are 2 mailboxes available?

How do you determine the sample space for the experiment where a coint and a xis sided die are tossed?

How to evaluate 8P7?

There are 12 paintings in a show. How Many ways can the paintings take first second or third?

How do you evaluate 100P3 using a calculator?