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?
Best Answer
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.