# Probability - Snow White and the 7 Dwarves

I'm stuck and I really need some direction as to how to tackle this problem.

Each morning, before they go off to work in the mines, the seven dwarves line up and Snow White kisses each dwarf on the top of his head. In order to avoid any hint of favoritism, she kisses them in random order each morning.

a. What is the probability that the dwarf named Bashful gets kissed first on Monday?

b. What is the probability that Bashful gets kissed first both Monday and Tuesday?

c. What is the probability that Bashful does not get kissed first, either Monday or Tuesday?

d. What is the probability that Bashful gets kissed first at least once during the week (Monday – Friday)?

e. What is the probability that, on Monday, Bashful gets kissed first and Grumpy second?

f. What is the probability, on Monday, that the seven dwarves will be kissed in perfect alphabetical order?

g. What is the probability that, on Monday, Bashful and Grumpy get kissed before any other dwarves?

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I am sure you have made some progress. It is useful if you indicate what calculations you have made. Then the answers can more directly address any problems you may have. – André Nicolas Oct 23 '12 at 15:10
You've edited back in the statistics tag, but I don't see any statistics in the problem. Do you? As for the problem itself, is it like any of the previous material in the course/text? Does any of this seem familiar? – AakashM Oct 23 '12 at 16:24
This is an expanded version of a homework problem from Problem Set 1 of ECE 313, Spring 2011, University of Illinois, and the solution is also on the same web site. The full version of the problem is likely to be found in the problem sets and solutions of other offerings of the same course. – Dilip Sarwate Oct 23 '12 at 19:44

These questions are conditional probabilities.

There is useful information in this regard on Wikipedia, and on Wolfram, but basically, conditional probabilities go like this:

$P(A|B)$ is the probability that event $A$ occurs given that event $B$ has already occurred. If you take the entire sample space, $S$, (the set of all possible events, or outcomes), then $A$ and $B$ represent sub-sets of that space.

Visually, you can think of these as Venn diagrams where $S$ is the entire area, and $A$ and $B$ are two smaller areas within the space. The intersecting, or shared space, between $A$ and $B$ is notated $AB$ (or $A\cap B$), while the combination of $A$ and $B$ is notated $A\cup B$.

Well, if we have that $B$ has occurred, and we want the probability that $A$ will occur, this must relate to the $AB$ sub-set, the set of events in which both $A$ and $B$ occur. We want to measure this relative to the initial probability that $B$ occurred, so:

$P(A|B) = \frac{P(AB)}{P(B)}$

Based on this, we can look at part (a) of the question.

a. What is the probability that the dwarf named Bashful gets kissed first on Monday?

Let's assign $A$ as the event that Bashful gets kissed first, and $B$ as the event that it is Monday. We are then looking for $P(A|B)$, the probability of Bashful being kissed first, given that it is Monday.

What is $P(AB)$? It is the probability that it is both Monday ($\frac{1}{5}$) and that Bashful is first to be kissed ($\frac{1}{7}$).

What is $P(B)$? It is the probability that it is Monday ($\frac{1}{5}$).

Therefore, $P(A|B)$ = $\frac{\frac{1}{5}\times \frac{1}{7}}{\frac{1}{5}}$ = $\frac{1}{7}$.

The remaining parts of the question follow the same process.

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This is more of a comment than an answer, so you know for when you get enough reputation to make comments. – Kevin Carlson Oct 23 '12 at 20:22
Thanks, @KevinCarlson, I've edited my answer to be more answer-like. – OrangeWombat Oct 24 '12 at 12:54