Probability denotes possibility. To be able to find the probability of a single event to occur, we need to know the total number of possible outcomes first.
Explore our app and discover over 50 million learning materials for free.
Lerne mit deinen Freunden und bleibe auf dem richtigen Kurs mit deinen persönlichen Lernstatistiken
Jetzt kostenlos anmeldenNie wieder prokastinieren mit unseren Lernerinnerungen.
Jetzt kostenlos anmeldenProbability denotes possibility. To be able to find the probability of a single event to occur, we need to know the total number of possible outcomes first.
Probability is the mathematical measure of how likely an event is to occur. It is the aspect of mathematics that deals with the occurrence of a random event.
Mathematically, probability ranges from 0 to 1, where 0 means it is impossible that the event will occur, and 1 means it is certain the event will occur. In between both values, we have varying degrees of likelihood of the event happening. 0.5 means there is an even chance of the event happening.
When a coin is tossed, for example, there are only two possibilities involved in the outcome. The outcome is either heads or tails. However, when you toss two coins, there are three possible outcomes involved. You either have two heads, two tails, or one head and one tail show up. We will now explore what formula is used to make these calculations, and to help us to solve more complex problems.
The formula for probability is defined as the possibility of an event happening is equal to the ratio of the number of favourable/desirable outcomes and the total number of outcomes. Mathematically, this is written as:
\[Probability \space of \space an \space event = \frac {Number \space of \space favourable \space outcomes}{Total \space number \space of \space outcomes}\]
Probability can be written down with the notation below:
P (A): the probability of event A happening.
P (A'): the probability of event A not happening.
Both situations should add up to 1. If event A has a probability between A happening and A not happening, then the probability of event A not happening \( = 1 - P (A')\). For example, if the \(P(A) = 0.75, \space then \space P(A') = 0.25\).
Assume there was an experiment where a dice was rolled 500 times, and you see 74 fives. The experiential probability of throwing a five is \(\frac{74}{500} = 0.148\).
Assuming the die is fair, then the theoretical probability of throwing a 5 is \(\frac{1}{6} = 0.166\).
This is because the 6 is the total possible outcomes there could be from throwing a dice. And 1 indicates the desired outcome (rolling a five).
So on average, if the die is fair, you will see \(\frac{1}{6} \space of \space 500 = 83\) fives.
This mathematics means that by throwing the dice 500 times, provided the dice is fair, you're going to roll approximately 83 fives.
When two coins are tossed simultaneously, what is the probability of getting heads on one coin and tails on the other?
Answer:
The sample space for when two coins are tossed is as follows;
S = {HH,HT,TH,TT}
Where H= Heads and T= Tails
\(Probability \space of \space an \space event = \frac {Number \space of \space favourable \space outcomes}{Total \space number \space of \space outcomes}\)
\(\begin{align}P(Head \space on \space one \space and \space Tail \space and \space on \space other) &= P(HT) + P(TH) \\ &= \frac {1}{4} + \frac {1}{4} \\ &= \frac {2}{4} \end{align}\)
Divide through by 2
\(P(Head \space on \space one \space and \space Tail \space on \space the \space other) = \frac {1}{2}\)
This means there is a 50% chance of having a head on one and a tail on another after tossing coins simultaneously.
What is the probability that a random card drawn from a pack of cards is a face card?
answer:
Since a standard deck has 52 cards, the total number of outcomes will be 52.
n(S) = 52
Now let E be the event of drawing a face card.
Number of favourable events = n (E)
\(n(E) = 3 \cdot 4 = 12\)
\(Probability = \frac{Number \space of \space Favourable \space Outcomes}{Total \space Number \space of \space Outcomes}\)
\(\begin{align} P(E) &= \frac{n(E)}{n(S)} \\ &= \frac {12}{52} \\ &=\frac{3}{13} \\ &=0.23 \end{align}\)
This can also be expressed in percentages as 23% probability.
We have a box that contains 4 blue balls, 5 red balls, and 11 white balls. If three balls get drawn from the vessel at random, what is the probability that the first ball is red, the second ball is blue, and the third ball is white?
Answer:
We will first find the probability for each colour picked.
Since there are 20 balls in all, the possible outcome for a pick is 20.
The probability that the first ball will be red then is \(\frac{5}{20}\).
We now have picked a ball which leaves us with the possible outcomes being 19.
So the probability of the second pick being blue is \(\frac{4}{19}\).
Again, because we have another pick already being made, our total possible outcomes reduce by 1, leaving us with 18.
The probability that the third ball is white is \(\frac{11}{18}\).
Therefore the entire probability that the first ball is red, the second ball is blue, and the third ball is white is \(\frac{5}{20} \cdot \frac {4}{19} \cdot \frac {11}{18} = \frac {44}{1368}\)
P = 0.032
This can also be expressed in percentages as 3.2% probability.
How to calculate probability: divide the number of events by the number of possible outcomes. This will give us the probability of a single event occurring.
How to calculate probability using Venn diagrams: List and group all possible events using set notation. After the listing is done, we use algebra or simple calculations to find the ratios or probabilities.
Probability = Number of favourable outcomes/ Total number of outcomes
How to find the probability of two events: just multiply the probability of the first event by the second. For example, if the probability of event A is 2/9 and the probability of event B is 3/9 then the probability of both events happening at the same time is (2/9)*(3/9) = 6/81 = 2/27.
What is probabilty?
Probability is the mathematical measure of how likely an event is to occur. It is the aspect of mathematics that deals with the occurrence of a random event.
What is the notation for probability of event A not happening?
P(A')
What is the notation for probability of event A happening?
P(A)
How many possible outcomes are there if a coin is tossed?
2
Both the probability of events happening and they not happening should add up to 1. Is this statement true or false?
True
What does it mean when the probability of an event happening is 0?
It means there is no probability that such an event will happen
Already have an account? Log in
Open in AppThe first learning app that truly has everything you need to ace your exams in one place
Sign up to highlight and take notes. It’s 100% free.
Save explanations to your personalised space and access them anytime, anywhere!
Sign up with Email Sign up with AppleBy signing up, you agree to the Terms and Conditions and the Privacy Policy of StudySmarter.
Already have an account? Log in
Already have an account? Log in
The first learning app that truly has everything you need to ace your exams in one place
Already have an account? Log in