A train has 20 seats of which 17 are broken. The conductor needs
to fill out a form that identifies the broken seats.
If she randomly identifes the 17 broken seats, what is the probability
that she identified all the correct seats? Express your answer in
scientific notation to 3 digit accuracy.
The question asks what is the probability that she identified all the correct seats.
To answer the question, you have to answer three sub questions.
In class, we use the shorthand ...
`Probability = \text( desired )/ \text( total )`
Of all the ways the conductor can mark the form, there's exactly one way that correctly
identifies the broken seats so the number of ways to get the desired outcome is `1`.
Thetotal number of outcomes is found by computing `( \ _20C_17)`
We use Choose instead of Permute because the order the seats are marked doesn't matter.
The answer is found by computing
`Probability = 1 / ( \ 20C_17)`
The answer in scientific notation is `8.77*10^(-4)`
Pro Tips!
When you calculate `\ _nC_r` or `\ _nP_r` , the correct answer is always a positive integer.
If your calculator gives you a negative or fractional answer, you've made a mistake.
Probability answers are always a fraction between 0 and 1 inclusive.That's because
the desired number of outcomes can't be bigger than the total number of outcomes.
Problem 2:
Antonella and Kai are taking one candy each. There are 4 Snickers and 15 Milky
Ways in a bag. If Antonella goes first and they choose randomly, what is the probability that they'll both
get a Snickers? Express your answer as a common fraction.
The question asks what is the probability that Antonella and Kai both get snickers?
The and is important as it tells you both events must happen. When you see an "and" in probablity,
that means you'll mutiply the probabilities together.
The problem has two events.
Antonella gets a snickers
Kai gets a snickers.
The probability that Antonella gets a snickers is 4/19 since there are 4 snickers
out of a total of 19 candies.
After Antonella has taken her snickers, there are 3 snickers left so the probability that
the event Kai gets a snickers is 3/18. 18 because there 18 candies left after Antonella took a snickers.
The compute the probabiLity that both events occur is found by:
`P_a & P_b = P_a * P_b`
where `P_a` is the probablity of the event Antonella gets a snicker and ` P_b` is
probablity of the event Kai gets a snickers .
Plug in the numbers:
`P_a & P_b = 4/19 * 3/18`
If you were doing the problem without a calculator, it would pay to notice the chance to divide before
multiplying. That makes the numbers smaller and easier to handle.
`P_a & P_b = 2/19 * 1/3`
`P_a & P_b =2/57`
Problem 3:
The top three steps of a stairway are shown. The stairway has 101 steps with 101 risers that are each 6 cm high. The 101 treads are
each 10 cm wide. If all the stairs were shown in the figure, what
would be the area of the figure?
The first riser forms a rectangle 6 cm high by 10 cm wide. Its area then is `6*10 = 60 cm^2`
Since there are 101 steps in the stairway, that means that there are:
`1 + 2 + 3 + 4 + ... + 101`
rectangles in the stairway. The series is an arithmetic series and the sum of an arithmetic series can be found by
`\sum = n*(F + L)/2`
where `F` represents the first number in the series, `L` represents the last number in the series and `n` represents
the number of terms in the series. Substituting we get
`\sum = 101*(1 + 101)/2`
`\sum = 101*( 102)/2`
`\sum = 101*51`
`\sum = 5151`
To get the total area then, multiply the number of rectangles by the area of a single rectangle, i.e.
`Area = 5151 * 60`
`Area = 309060`
Problem 4:
Compute f(13) where f(n) is as shown.
Express your answer in scientific notation
to 4 digit accuracy.
This is a calculator question for you to practice entering expressions properly.
You are asked to compute f(13). That means everywhere there's an 'n', you substitute a '13' like so:
`f(13) = ((1+\sqrt(5))^13 - (1-\sqrt(5))^13)/(2^13\sqrt(5))`
Enter everything on the right side of the equation into your calculator and verify it looks like the expression shown. When it does, press the `=` key.
You should see a number like 233 or `2.3 * 10^2`.
Click Setup to adjust the format to scientific notation and set the precison to 4 digit accuracy and you should see `2.330*10^2` in the display.
Problem 5:
What is the single discount that is equivalent to the two
successive discounts of 41 % off followed by 27% off the
discounted _price? Answer to the nearest hundredth of a percent.
You can pretend the price is $100 and apply the discounts. The number you have at the end is the
price you would pay after the discounts. The single discount then would be
`100 - (\text(price you would pay))`
It doesn't matter what price you choose to start with.The single discount is found by computing
`\text((the price you started with)) - ` `(\text(the price you would pay after applying the discounts to the starting price)) `
We'll compute the answer by assuming the starting price is $100.
A 41% discount on $100 is $41 so we have a new price of $59.
We then compute the second discount:
`$59*27/100 = $15.93`
`$59-$15.93 = $43.07`
$43.07 is the price after the two discounts so the single discount that would give you a final price of $43.07 if you started at $100 is
`$100 - $43.07 `
or 56.93% Notice the "%" follows the number.
