Independent events
You toss a coin and roll a die at the same time. You draw a card, put it back in the pack and draw another. Two machines in a factory check the same part separately. In all three cases, what happens in one part of the experiment does not change anything about what can happen in the other: they are independent events.
What it means for two events to be independent
A fair coin is flipped and a standard cubic die (with faces numbered from 1 to 6) is rolled. What is the probability of obtaining "Tails" on the coin and a "5" on the…
A fair coin is flipped and a standard cubic die (with faces numbered from 1 to 6) is rolled. What is the probability of obtaining "Tails" on the coin and a "5" on the die?
Two events $A$ and $B$ of a random experiment satisfy $P(A) = 0,4$, $P(B) = 0,5$ and $P(A \cap B) = 0,2$. Are events $A$ and $B$ independent? Justify your answer…
Two events $A$ and $B$ of a random experiment satisfy $P(A) = 0,4$, $P(B) = 0,5$ and $P(A \cap B) = 0,2$. Are events $A$ and $B$ independent? Justify your answer mathematically.
The product rule
When you draw with replacement (what you drew is put back before the next draw), the make-up does not change: the draws are independent.
In an urn there are 3 red balls and 4 blue balls. A ball is drawn, its colour recorded, returned to the urn and a second ball is drawn. Calculate the probability that…
In an urn there are 3 red balls and 4 blue balls. A ball is drawn, its colour recorded, returned to the urn and a second ball is drawn. Calculate the probability that the first ball is red and the second is blue.
From a Spanish deck of 40 cards, a card is drawn, its value recorded, replaced in the deck and a second card is drawn. Find the probability of drawing a "King" on the…
From a Spanish deck of 40 cards, a card is drawn, its value recorded, replaced in the deck and a second card is drawn. Find the probability of drawing a "King" on the first draw and a card of the suit of "Cups" on the second.
Several independent events at once
In systems with several components that fail separately (servers, alarms, engines), the product rule is applied as many times as there are components.
A student answers a short 3-question test completely at random. Each question has 4 answer options and only one is correct. What is the probability of answering all the…
A student answers a short 3-question test completely at random. Each question has 4 answer options and only one is correct. What is the probability of answering all the questions correctly?
A hospital computer system has 3 independent servers to prevent service downtime. The probability of a server failing on any given day is $0,05$. What is the probability…
A hospital computer system has 3 independent servers to prevent service downtime. The probability of a server failing on any given day is $0,05$. What is the probability that the system keeps working (that is, at least one server does not fail) for an entire day?