\item How many ways are there to choose two animals and three plants?
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\item The manager finds that he only has 5 exhibits available. How many different sets of animals can he choose, if only one can be in each exhibit time?
A \textit{probability}, also known as a \textit{chance}, is a number showing how likely some event is to happen. Let us call the event $X$. Then the probability of $X$ taking place is
In some of the following problems, we will be flipping a coin. Let us use $H$ to represent the event of the coin landing heads, and $T$, the event of the coin landing tails.
Some parts of the previous problem involve repeated trials: two dice, or two coins. You may have solved these by listing out all the possible outcomes. Though this simple approach works for small problems, it isn't particularly useful for larger ones: ten coin flips create 1024 possible outcomes, and ten dice rolls, 60466174.
A better way to think about repeated trials is as a ``tree,'' where each outcome represents a path. The following tree represents two coin flips, and the four paths down it (from left to right) correspond to the four possible outcomes: HH, HT, TH, TT.
If we have two independent events $A$ and $B$, then $\Pt{A and B}=\P{A}\times\P{B}$. \\
Usually we write $\Pt{A and B}$ as $\P{A \cap B}$. \\
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Here's another important thought:
\proposition{}
If the probability of event $A$ happening is $\P{A}$, the probability of $A$\textit{not} happening is $1-\P{A}$
\pagebreak
\problem{}
There are three cans of white paint and two cans of black paint in a dark storage room. You take two cans out without looking. What is the probability that you'll choose two cans of the same color?
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\problem{}
Hospital records show that of patients
suffering from a certain disease,
75\% die of it. What is the probability
that of 5 randomly selected patients,
4 will recover? \\
\hint{What is the probability of a patient recovering?}
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\problem{}
When Oleg calls his daughter Anya,
the chance of the call getting through is 60\%.
How likely is it to have at least one connection
in four calls?
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\problem{}
The chance of a runner to improve
his own personal record in a race is $p$.
What is the probability that his record will improve after 3 races?
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\newpage
\problem{}
You toss a pair of fair dice five times.
What is the probability that you get a sum of ten exactly two times?
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\problem{}
You toss a pair of fair dice five times.
What is the probability that you get ten
at least twice?
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\problem{}
A fair coin is tossed 4 times. What is the chance of getting more heads than tails?
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\problem{}
A pharmaceutical study shows that a new drug causes negative side effects in 3 of every 100 patients.
To check the number, a researcher chooses 5 random people to survey.
