The probability of complementary events refers to the probability associated with events not occurring. For example, if
, then the probability of
not occurring is the probability associated with all other events in
occurring less the probability of
occurring. This means that
where A' refers to `not A'
In other words, the probability of `not A' is equal to one minus the probability of A.
If you throw two dice, one red and one blue, what is the probability that at least one of them will be a six?
To solve that kind of question, work out the probability that there will be no six.
The probability that the red dice will not be a six is 5/6, and that the blue one will not be a six is also 5/6.
So the probability that neither will be a six is
.
So the probability that at least one will be a six is
.
A bag contains three red balls, five white balls, two green balls and four blue balls:
1. Calculate the probability that a red ball will be drawn from the bag.
2. Calculate the probability that a ball which is not red will be drawn
Let R be the event that a red ball is drawn:
P(R)-n(R)/n(S)=3/14
R and R' are complementary events
P(R') = 1 - P(R) = 1 -3/14 = 11/14
Alternately P(R') = P(B) + P(W) + P(G)
P(R') = 4/14 + 5/14 + 2/14 = 11/14
Interpretation of probability values
The probability of an event is generally represented as a real number between 0 and 1, inclusive. An impossible event has a probability of exactly 0, and a certain event has a probability of 1, but the converses are not always true: probability 0 events are not always impossible, nor probability 1 events certain. The rather subtle distinction between "certain" and "probability 1" is treated at greater length in the article on "almost surely".
Most probabilities that occur in practice are numbers between 0 and 1, indicating the event's position on the continuum between impossibility and certainty. The closer an event's probability is to 1, the more likely it is to occur.
For example, if two mutually exclusive events are assumed equally probable, such as a flipped or spun coin landing heads-up or tails-up, we can express the probability of each event as "1 in 2", or, equivalently, "50%" or "1/2".
Probabilities are equivalently expressed as odds, which is the ratio of the probability of one event to the probability of all other events. The odds of heads-up, for the tossed/spun coin, are (1/2)/(1 - 1/2), which is equal to 1/1. This is expressed as "1 to 1 odds" and often written "1:1".
Odds a:b for some event are equivalent to probability a/(a+b). For example, 1:1 odds are equivalent to probability 1/2, and 3:2 odds are equivalent to probability 3/5.
Summary
Random experiments
Outcome
Sample space
Event
Probability models
Classical theory
Relative frequency vs. probability
Probability identities
Mutually exclusive events
Complementary events
End of chapter exercises
A group of 45 children were asked if they eat Frosties and/or Strawberry Pops. 31 eat both and 6 eat only Frosties. What is the probability that a child chosen at random will eat only Strawberry Pops?
In a group of 42 pupils, all but 3 had a packet of chips or a Fanta or both. If 23 had a packet of chips and 7 of these also had a Fanta, what is the probability that one pupil chosen at random has:
Both chips and Fanta
has only Fanta?
Use a Venn diagram to work out the following probabilities from a die being rolled:
A multiple of 5 and an odd number
a number that is neither a multiple of 5 nor an odd number
a number which is not a multiple of 5, but is odd.
A packet has yellow and pink sweets. The probability of taking out a pink sweet is 7/12.
What is the probability of taking out a yellow sweet
If 44 if the sweets are yellow, how many sweets are pink?
In a car park with 300 cars, there are 190 Opals. What is the probability that the first car to leave the car park is:
an Opal
not an Opal
Tamara has 18 loose socks in a drawer. Eight of these are orange and two are pink. Calculate the probability that the first sock taken out at random is:
Orange
not orange
pink
not pink
orange or pink
not orange or pink
A plate contains 9 shortbread cookies, 4 ginger biscuits, 11 chocolate chip cookies and 18 Jambos. If a biscuit is selected at random, what is the probability that:
it is either a ginger biscuit of a Jambo?
it is NOT a shortbread cookie.
280 tickets were sold at a raffle. Ingrid bought 15 tickets. What is the probability that Ingrid:
Wins the prize
Does not win the prize?
The children in a nursery school were classified by hair and eye colour. 44 had red hair and not brown eyes, 14 had brown eyes and red hair, 5 had brown eyes but not red hair and 40 did not have brown eyes or red hair.
How many children were in the school
What is the probility that a child chosen at random has:
Brown eyes
Red hair
A child with brown eyes is chosen randomly. What is the probability that this child will have red hair
A jar has purple, blue and black sweets in it. The probability that a sweet, chosen at random, will be purple is 1/7 and the probability that it will be black is 3/5.
If I choose a sweet at random what is the probability that it will be:
purple or blue
Black
purple
If there are 70 sweets in the jar how many purple ones are there?
1/4 if the purple sweets in b) have streaks on them and rest do not. How many purple sweets have streaks?
For each of the following, draw a Venn diagram to represent the situation and find an example to illustrate the situation.
A sample space in which there are two events that are not mutually exclusive
A sample space in which there are two events that are complementary.
Use a Venn diagram to prove that the probability of either event A or B occuring is given by: (A and B are not exclusive)
P(A or B) = P(A) + P(B) - P(A and B)
All the clubs are taken out of a pack of cards. The remaining cards are then shuffled and one card chosen. After being chosen, the card is replaced before the next card is chosen.
What is the sample space?
Find a set to represent the event, P, of drawing a picture card.
Find a set for the event, N, of drawing a numbered card.
Represent the above events in a Venn diagram
What description of the sets P and N is suitable? (Hint: Find any elements of P in N and N in P.)
Thuli has a bag containing five orange, three purple and seven pink blocks. The bag is shaken and a block is withdrawn. The colour of the block is noted and the block is replaced.
What is the sample space for this experiment?
What is the set describing the event of drawing a pink block, P?
Write down a set, O or B, to represent the event of drawing either a orange or a purple block.
Draw a Venn diagram to show the above information.
Questions & Answers
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
