Conditional Probability

Key Questions

• How is Bayes' Theorem used to solve complex probability questions?

  Answer:

  Using the ex-post probabilities, it tries to calculate ex-ante probability.

  Explanation:

  Suppose there are 3 production units A, B and C. They are used to produce Good-X. Again, the probabilities of a randomly selected item from A, B and C are known.

  If we mixed up all their output. Now, if we select one item, we do not know from which production unit, it has come from.

  O.K., Suppose the selected Item is defective, we can find out the probability about which production unit has produced it.

  Nallasivam V · 1 · Aug 14 2015
• How does the General Multiplication rule differ from the Special Multiplication Rule of Probability?

  General Multiplication Rule of Probability is related to a probability of a combined occurrence of any two events #A# and #B# (denoted as #A*B#) expressed in term of their individual (denoted as #P(A)# and #P(B)# correspondingly) and conditional probabilities (probability of occurrence of one event under condition of occurrence of another, denoted as #P(A|B)# and #P(B|A)# correspondingly):

  #P(A*B)=P(A)*P(B|A)=P(B)*P(A|B)#

  Special Multiplication Rule is related to a probability of a combined occurrence of two independent events (that is, the probability of one is not dependent on the probability of another or, in other words, conditional probability of one under condition of occurrence of another equals to its unconditional probability). This necessitates #P(A|B)=P(A)# and #P(B|A)=P(B)#, and the multiplication rule looks like this:

  #P(A*B)=P(A)*P(B)#

  Informative lectures and solutions to many problems of the Theory of Probabilities for beginners can be found in the corresponding chapter on the Web site Unizor (free online course of advanced mathematics for teenagers). There you can find definitions and properties of main objects the Theory of Probabilities deals with - sample space, events, measures of probability, independence etc.

  Zor Shekhtman · 1 · Mar 1 2015

• The probability that you are late to school is 0.05 for any day. Given that you slept late, the probability that you are late to school is 0.13. Are the events "Late to School" and "Slept Late" independent or dependent?
• What is the meaning of conditional probability?
• A card is drawn from a standard deck. A second card is drawn, without replacing the first card. Consider the following events: Event A = The first card selected is red. Event B = The second card selected is black. What word describes the relationship between events A and B?
• A card is drawn from a standard deck. A second card is drawn, without replacing the first card. What is the probability that the first card is red and the second card is black?
• A card is drawn from a standard deck, its color is noted, and then it is replaced in the deck. Another card is then drawn from the deck. What is the probability that the first card is red and the second card is black?
• How does the General Multiplication rule differ from the Special Multiplication Rule of Probability?
• How is Bayes' Theorem used to solve complex probability questions?
• If the letters of the word Mississippi are placed in a hat, what is the probability that the first two letters drawn are both "i" if the first letter drawn is not replaced?
• In a two-child family, if one child is a boy, what is the probability that the other child is a girl? If the first child is a boy, what is the probability that the second is a girl?
• In a two child family, if one child is a boy, what is the probability that the other child is a girl? If the first child is a boy, what is the probability that the second is a girl?
• If you roll two die, what is the probability that the sum of the die rolls is 6? What is the probability that the sum is 6 if one die rolls a 4?
• Assume that A and B are events in a sample space and that P(A)=.40 and P(B/A) = .25. What is the probability that both A & B occur?
• A survey of farmers showed that last year 15% bought a new tractor, 21% of them bought a new car, and 8% of them bought both a new tractor and a new car. What is the probability that a farmer bought a new tractor, given that he bought a new car?
• A bag contains 9 red marbles and 3 green marbles. How can you find the probability of drawing first one green marble and then one red marble if you do not replace the first marble?
• What is the conditional probability of an event and how does it differ from the regular probability of an event occurring?
• 3 balls are drawn from a jar that contains 5 white balls and 3 black balls. What is the probability that all 3 balls are same in color if there is no replacement?
• What is the probability that the sum of two dice rolls is 12 if the sum is even?
• Of 100 gumdrops 73 are large, 39 of them are red, and 19 are large and red. If you choose a random gumdrop, what is the probability that it is red? If the gumdrop is red, what is the probability that it is large?
• A box of light bulbs has 6 75W bulbs, 4 40W bulbs, and 5 60W bulbs. If 2 bulbs are taken from the box and one is a 75W bulb, what is the probability that both are 75W? If neither bulb is 75W what is the probability that both have the same wattage?
• In a game of single deck blackjack, what is the likelihood of being dealt a blackjack if three other hands, each with an ace, are also dealt?
• What does it mean if P(A|B) = P(A) and P(B|A) - P(B)?
• What is the definition of conditional probability?
• What is the Kolmogorov formula for conditional probability?
• If A and B are mutually exclusive, what is P(A|B)?
• What is the mathematical definition of conditional probability?
• If A and B are two independent random variables, is P(A|B)<P(A)?
• What is Bayes' theorem?
• What is a Bayesian inference?
• What is Bayes' rule?
• How do you apply Bayes' theorem to continuous random variables?
• What general form of the mass function of a conditional probability distribution for discrete random variables?
• If P(A) = 0.1, P(B) = 0.9, and P(A#nn#B)=0, what is P(B|A)? What about P(A|B)?
• If P(A) = 0.8, P(B) = 0.9, and P(A#nn#B)=.8, what is P(B|A)? What about P(A|B)?
• If #P(A) = 0.7, P(B) = 0.9# and #P(A nn B)=.4#. What is #P(B|A)#? What about #P(A|B)#?
• If P(A) = 0.7, P(B) = 0.1, and P(A#nn#B)=0, what is P(B|A)? What about P(A|B)?
• If P(B|A) = 0.6 and P(A) = 0.5, what is P(B)?
• The probability of event X is 0.4. If events X and Y are complements, what is the P(Y)?
• What does it mean about events #A# and #B# if #P(A|B)=P(B|A)#?
• Ron draws 16 cards from a standard deck of 52. The deck is made up of equal numbers of four suits clubs diamonds, hearts, and spades. How many of the cards drawn can Ron expect to be spades?
• Question #5a6e7
• Question #5170b
• Question #7d4be
• A bowl contains 7 pennies, 9 nickels, and 4 dimes. Elyse removes one coin at random from the bowl and does not replace it. She then removes a second coin at random. What is the probability that both will be dimes?
• Question #59f14
• Conditional probability question?
• A family has 6 children. I've figured out that there are 64 different ways the genders of the kids can work out. What's the probability that there's at least 1 boy in the family?
• Question #a7582
• Which of the following statements is a conditional statement?
• The probability of a lady over 40 years of age not having had children is 0.1. In a sample of 60 ladies over 40, what are the chances that between 5 and 9 of them have had no children?
• Two events A and B are such that #P(A)=.2, P(B)=.3,# and #P(AuuB)=.4#. What is #P(AnnB)#?
• 90 students will graduate from Lima Shawness High School this year. Out of the 90, 50 are planning to attend college. Two students are selected at to carry the flag at graduation. What is the probability that both of them are planning to attend college?
• There are two boxes - one holds 11 cards numbered 1 through 11, the other 5 cards numbered 1 through 5. Two cards are randomly drawn. What is the probability of drawing 2 even cards in a row?
• Assume that A and B are events in a sample space and that P(A) = .40 and P(B|A) = .25. How do you find P(A intersection B)?
• A factory has three machines, A, B, and C, for producing items. Machine A produces 50%, B produces 30%, and C produces 20%. If a randomly selected item from the factory's output is found to be defective, what is the probability that B made it?
• Batches of serum are processed by three different departments having rejection rates of 0.10, 0.08, and 0.12 respectively. What is the probability that a batch of serum survives the first departmental inspection but is rejected by the second department?
• I have two servers at a grocery store. P(server 1) is 0.4. P(server 2) is 0.6. P(time is less than 5 min | server 1) is 1. P(time is less than 5 min | server 2) is 4/9. What is P(server 1 | time is less than 5 min)?
• Two fair dice (one red and one green) are rolled. What is the probability that the sum is 5, given that the green one is either 4 or 3?
• The probability of having a toothache because of a cavity is 0.8. Suppose the probability of having a cavity is 0.05. Assuming the probability of a toothache given you have no cavity is 0.01, what's 'the probability of a cavity if you have a toothache?
• Among the contestants in a competition are 42 women and 28 men. If 5 winners are randomly selected, what is the probability that they are all men?
• In a string of 12 christmas lights, 3 are defective. Bulbs are selected at random, one at a time, until the third defective bulb is found. What is the probability that the third defective bulb is the third bulb tested?
• #P(E)=.24# and #P(F)=.3#. What is #P(E uu F)#?
• On average a shop sells 750 kg per day with a standard deviation of 60 kilo’s. We assume sales are normal. If the shop has 850 kilograms in stock at the beginning of the day, what is the probability it runs out that day?
• The probability that a married woman watches a certain television show is 0.4, and the probability that her husband watches the show is 0.5. What's the probability both the husband and wife watch the show?
• In a certain community, 25% of the families own a dog, and 20% of the families that own a dog also own a cat. It is also know that 28% of all the families own a cat. What is the probability that a randomly selected family owns a cat?
• A card is drawn from a shuffled deck of 52 cards, and not replaced. Then a second card is drawn. What is the probability that the second card is a king?
• Tenzin's friends assure him that if he asks Mikala out on a date, there is an 85% chance that she will say yes. If there is a 60% chance that Tenzin will summon the courage to ask Mikala out to the dance next week, what are the odds she will say yes?
• In a two-child family, one child is a boy. What is the probability that the other child is a girl?
• Is there a formula for such a multiple conditional probability?
• Choose 2 cards from a standard 52 card card, in succession and without replacement. What is the probability that the second card is a king given that the first card is a face card?
• Assume 75% of the AP stats students studied for this test. If 40% of those who studied get an A but only 10% of those who did not study get A, what is the probability that someone who gets an A actually studied for the test?
• A die is tossed. What is P(of the number being prime knowing that its even)?
• Two coins are tossed. Given that the first is a head, what is the probability of getting another head?
• 75% students passed math, 85% passed chemistry, and 90% passed math or chemistry. A student is selected at random. What is the probability that the student passed mathematics, given that the student passed chemistry?
• One bag contains 4 white balls and 6 black balls. Another bag contains 8 white balls and 2 black balls. A coin is tossed to select a bag, then a ball is randomly selected from that bag. What is the probability that a white ball will be drawn?
• What is the conditional probability that a card drawn at random from a pack of 52 cards is a face card, given that the drawn card is a spade?
• Suppose that P(A) = 0.3 and P(B) = 0.25 and P(A ∩ B) = 0.1. What is P(B | A(complement))?
• When drawing two hearts from a deck without replacement, are the events independent?
• What is the value of #n(AnnB)# if #n (A)= 7, n(B)= 9#, and #n(AuuB)=13#?
• A bag contains 4 red, 3 yellow and 2 purple discs. A disc is taken, at random, from the bag and is not replaced. A second disc is then taken, at random, from the bag. Calculate the probability that the two discs taken from the bag are different colours?
• Question #1c547
• A restaurant offers the possibility of 168 three-course dinners. Each dinner has an appetizer, an entrée, and a dessert. If the number of appetizers decreases from 7 to 5, how many fewer possible three-course dinners can the restaurant offers?
• If #"P"(Q) = 4/7# and #"P"(R) = 1/2#, and Q and R are independent events, then what is #"P"(Q nn R)#?
• Question #d5175
• We open a 1500 page book to a random page. What is the probability that we open to?
• From a standard deck, what are the probabilities of A. picking a red card, B. picking a spade, C. picking an Ace?
• What is the probability that the last three digits of a randomly selected phone number are all prime?
• Suppose A and B are independent events. If #P(A) = .4# and #P(B) = .1#, what is #P(AnnB)#?
• Question #bb618
• What is the probability that in three consecutive rolls of two fair dice, a person gets a total of 7, followed by a total of 11, followed by a total of 7?
• Please answer the following question?
• Please solve the question by Bayes theorem?
• You randomly choose a sock from a drawer of 8 black socks, 10 white socks, 12 blue socks, and 6 green socks. Then, without replacing the sock, you randomly choose a second sock. Find the probability that you choose 2 blue socks?