6.2: Basic Concepts
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If you roll a die, pick a card from deck of playing cards, or randomly select a person and observe their hair color, we are executing an experiment or procedure. In probability, we look at the likelihood of different outcomes. We begin with some terminology.
The result of an experiment is called an outcome.
An event is any particular outcome or group of outcomes.
A simple event is an event that cannot be broken down further.
The sample space is the set of all possible outcomes.
If we roll a standard 6-sided die, describe the sample space and some simple events.
Solution
Some examples of simple events:
- We roll a 1
- We roll a 5
Some examples of events:
- We roll a number bigger than 4
Given that all outcomes are equally likely, we can compute the probability of an event \(E\) using this formula:
\[
P(E)=\dfrac{\text { Number of outcomes corresponding to the event } \mathrm{E}}{\text { Total number of equally likely outcomes }}
\]
If we roll a 6-sided die, calculate
a) P(rolling a 1)
b) P(rolling a number bigger than 4)
Solution
Recall that the sample space is \(\{1,2,3,4,5,6\}\)
a) There is one outcome corresponding to "rolling a 1 ", so the probability is \(\dfrac{1}{6}\)
b) There are two outcomes bigger than a 4 , so the probability is \(\dfrac{2}{6}=\dfrac{1}{3}\)
If we roll a 6-sided die, calculate P(rolling an odd number).
- Answer
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P(rolling an odd) = 3/6 = 1/2
Probabilities are essentially fractions, and can be reduced to lower terms like fractions.
Let's say you have a bag with 20 cherries, 14 sweet and 6 sour. If you pick a cherry at random, what is the probability that it will be sweet?
Solution
There are 20 possible cherries that could be picked, so the number of possible outcomes is 20 . Of these 20 possible outcomes, 14 are favorable (sweet), so the probability that the cherry will be sweet is \(\dfrac{14}{20}=\dfrac{7}{10}\).
There is one potential complication to this example, however. It must be assumed that the probability of picking any of the cherries is the same as the probability of picking any other. This wouldn't be true if (let us imagine) the sweet cherries are smaller than the sour ones. (The sour cherries would come to hand more readily when you sampled from the bag.) Let us keep in mind, therefore, that when we assess probabilities in terms of the ratio of favorable to all potential cases, we rely heavily on the assumption of equal probability for all outcomes.
At some random moment, you look at your clock and note the minutes reading.
a. What is probability the minutes reading is 15?
b. What is the probability the minutes reading is 15 or less?
- Answer
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a. 1/60 b. 16/60
A standard deck of 52 playing cards consists of four suits (hearts, spades, diamonds and clubs). Spades and clubs are black while hearts and diamonds are red. Each suit contains 13 cards, each of a different rank: an Ace (which in many games functions as both a low card and a high card), cards numbered 2 through 10, a Jack, a Queen and a King. The Jack, Queen, and King are called face cards.
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Compute the probability of randomly drawing one card from a deck and getting an Ace.
Solution
There are 52 cards in the deck and 4 Aces so \( P( Ace)=\dfrac{4}{52}=\dfrac{1}{13} \approx 0.0769\)
We can also think of probabilities as percents: There is a \(7.69 \%\) chance that a randomly selected card will be an Ace.
Notice that the smallest possible probability is 0 (there are no outcomes that correspond with the event). The largest possible probability is 1 (all possible outcomes correspond
with the event).
An impossible event has a probability of 0 .
A certain event has a probability of 1 .
The probability of any event must be \(0 \leq P(E) \leq 1\)
In the course of this chapter, if you compute a probability and get an answer that is negative or greater than 1, you have made a mistake and should check your work.
Find the probability of drawing a single card from a deck and getting a red spade.
Solution
There are 52 cards in the deck, but all spades are black. It is impossible to draw a single card and have that card be a red spade. This is an impossible event, so
\[ \mathbb{P}(\) red spade \()=\dfrac{0}{52}=0\].
You wake up on a cloudy, drizzly Seattle day to the weatherman saying the chance of rain today is 100%. Find the probability that it will rain today.
Solution
Since we are told the chance of rain is 100%, we know that this is a certain event.
\[ \mathbb{P}(rain)=\dfrac{100}{100}=1\].
Your neighbor tells you that he heard on the radio that the probability of an earthquake
of magnitude 7.0 or higher hitting California in the next 10 years is 1.2. How do you
know that this is incorrect?
- Answer
-
Probabilities must be between 0 and 1, so a probability of 1.2 is impossible.