The Total Probability Formula. Bayes' Formula

Two balls are drawn at random from an urn containing 3 white and 2 black balls, and are placed into a second urn containing 4 white and 4 black balls. Find the probability of then drawing a white ball from the second urn.

A student came to an examination knowing only 24 out of 30 questions. If the student does not know the answer to the question asked, the instructor asks another, additional question. The student passes if they answer at least one question correctly. What is the probability that the student passes?

Suppose there are $n$ identical urns. Urn number $i$ is known to contain $M_i$ balls, of which $m_i$ are white. An urn is chosen at random, and a ball is drawn from it. What is the probability that the ball drawn is white?

Among $N$ examination tickets, $n$ are lucky.'' Students come up one after another to take a ticket. Who has the greater probability of drawing a lucky'' ticket: the one who goes first, or the one who goes second? Or the one who goes third?

The probability that a camouflaged enemy is located in the area under fire is $3/10$; the probability of hitting him in this case on each individual shot is $1/5$. What is the probability of a hit:

Three players A, B, and C take part in a game under the following scheme. In the first round, A and B play, while C sits out. The loser is replaced by player C, and in the second round the winner plays against C, while the player defeated in the first round sits out. The competition continues in this fashion until one of the players wins two games in a row, at which point that player is declared the winner. Assuming the players are equally skilled and excluding the possibility of draws, find the probability that:

Sports clubs $A$ and $B$ compete with three teams each. The probabilities of $A$'s teams winning their matches against the corresponding teams of $B$ may be taken as $4/5$ for the first team (against $B$'s first team), $2/5$ for the second (against $B$'s second team), and $2/5$ for the third (against $B$'s third team). Winning requires taking at least two of the three matches (there are no draws). Whose victory is more likely?

A box contains $a$ new and $b$ used tennis balls. Two balls are drawn from the box at random and played with. Afterward the balls are returned to the box. Some time later, two balls are again drawn from the box at random. Find the probability that they are new.

Suppose that on average 5 men out of 100 and 25 women out of 10000 are colorblind. A randomly chosen person is found to be colorblind. What is the probability that this person is a man?

There are 5 urns with the following composition: the first and second urns each contain 2 white and 3 black balls; the third and fourth urns each contain 1 white and 4 black balls; the fifth urn contains 4 white and 1 black ball. A ball is drawn from a randomly chosen urn. It turns out to be:

At a bolt factory, machines $A$, $B$, and $C$ account for 25, 35, and 40 of all output, respectively. Among their output, the proportion of defective bolts is 5, 4, and 2, respectively. A bolt chosen at random from the output turns out to be defective. What is the probability that it was manufactured:

A certain item is manufactured by two factories, with the second factory's output being $k$ times that of the first. The proportion of defectives at the first factory is $p_1$, and at the second, $p_2$. Items produced by the two factories over the same period of time are mixed together and put on sale. What is the probability that an item you bought came from the second factory, given that it turned out to be defective?

From an urn containing $m \geqslant 3$ white balls and $n$ black balls, one ball of unknown color was lost. To determine the composition of the balls in the urn, two balls are drawn from it at random. Find the probability that the ball lost was white, given that the balls drawn turned out to be white.

Under certain firing conditions, marksman $A$ hits the target with probability $p_1 = 3/5$, marksman $B$ with probability $p_2 = 1/2$, and marksman $C$ with probability $p_3 = 2/5$. The marksmen fired a volley at the target, and two bullets hit it. Which is more likely, that $C$ hit the target or that they did not?

Over a communication channel, one of three letter sequences may be transmitted: $AAAA$, $BBBB$, $CCCC$, with prior probabilities $3/10$, $2/5$, and $3/10$, respectively. It is known that, due to noise, the probability of correctly receiving any given letter is $3/5$, while the probabilities of that letter being turned into any other specific letter are equal and each equal to $1/5$. The letters are assumed to be distorted independently of one another. Find the probability that the sequence $AAAA$ was transmitted, given that the sequence $ABCA$ was received.

The first gun of a two-gun battery is sighted so that its probability of a hit is $3/10$, while for the second gun it is $1/5$. The battery fired two salvos, with one of the guns firing only a single shot; one hit and two misses were recorded. What is the probability that it was the first gun that fired the single shot?

Items from a certain production process meet the standard with probability $0{,}96$. A simplified testing system is proposed, which gives a positive result with probability $0{,}98$ for items that meet the standard, and with probability $0{,}05$ for items that do not meet the standard. What is the probability that:

Four faces of die $A$ are red and two are white; die $B$ has two red faces and four white ones. A coin is tossed once. If heads comes up, only die $A$ is rolled from then on; if tails, only die $B$.

Consider an electrical circuit made up of relays according to the following scheme:

[Figure omitted -- see source PDF page 50: a bridge circuit with five relays $A$, $B$, $C$, $D$, $E$ --- input splits into two branches, the upper branch through $A$ then $B$ and the lower branch through $C$ then $D$, with $E$ connecting the midpoint between $A,B$ to the midpoint between $C,D$, recombining into the output]

Each of the relays $A$, $B$, $C$, $D$, and $E$, operating independently, is open or closed with probabilities $p$ and $1-p$ respectively.

Three urns contain white and black balls: the first urn has 2 white and 3 black balls, the second has 2 white and 2 black balls, the third has 3 white and 1 black ball. A ball is drawn at random from the first urn and transferred to the second. Then a ball is drawn at random from the second urn and transferred to the third. Finally, a ball is transferred from the third urn to the first.

An urn contains $m$ white and $n$ black balls. A ball is drawn at random; it is returned to the urn, and, in addition, $k$ balls of the same color as the one drawn are added to the urn. What is the probability that the second ball drawn will be white?

An urn contains 12 balls, of which 8 are black and 4 are white. Three players $A$, $B$, and $C$ draw balls in turn. Whoever first draws a white ball wins. Estimate each player's chances of success.

$N$ marksmen fire at a single target in turn. The probability that the $i$-th marksman hits the target is $p_i$. The marksman who is first to hit the target is declared the winner. Each marksman has $n$ cartridges. Determine the probability that the $k$-th marksman wins.

$A$ tells the truth in 3 cases out of 4, and $B$ in 4 cases out of 5. From an urn containing 9 balls of various colors, including one white ball, a single ball was drawn. $A$ and $B$ both looked at it and both said that the ball was white. Find the probability that they told the truth.

Missiles are fired at an observed target. The probability that each missile hits the target is $p$; hits of individual missiles are independent. Each missile that hits destroys the target with probability $p_0$. Firing continues until the target is destroyed or the entire supply of missiles is exhausted; the base has $n$, $n > 2$, missiles. Find the probability that not all the missiles will be expended.

Suppose the probability of hitting a target with one shot is $p$, and the probability that the target is destroyed given $k$ hits on it is $1 - r^k$. What is the probability that the target is destroyed if $n$ shots are fired?

Suppose an insect lays $r$ eggs with probability $\dfrac {\lambda^r}{r!} e^{-\lambda }$. The probability that an insect develops from an egg is $p$. Assuming the insects develop independently of one another, find the probability that $k$ new insects emerge from a single clutch of eggs.

A submarine attacks a ship, firing $n$ torpedoes at it in succession and independently. Each torpedo hits the ship with probability $p$ and, if it hits, is equally likely to hit any one of the $k$ compartments into which the ship's underwater hull is divided. A torpedo that hits a compartment causes it to flood. The ship sinks if at least two compartments are flooded. What is the probability that the ship will be sunk?

$n$ independent shots with incendiary shells are fired at a fuel tank. Each shell hits the tank with probability $p$. If one shell hits the tank, the fuel ignites with probability $p_0$, while if two shells hit it, it ignites with certainty. Find the probability that the fuel will ignite after $n$ shots.

Three guns fire at three targets. For each gun, a target is chosen at random and independently of the others. A target fired at by one gun is destroyed with probability $p$. What is the probability that exactly two of the three targets will be destroyed?

Suppose the number of offspring of a certain insect is random and has a geometric distribution with parameter $p$. Each offspring is equally likely to be male or female. Find the probability that there will be $k$ males among the offspring.

Hint. Use the equality

Two players $A$ and $B$, having capitals $a$ and $b$ respectively, play a game of chance consisting of individual rounds. Suppose that in each round player $A$ wins with probability $p$ and loses with probability $1-p$. After each round, the loser pays 1 ruble to the winner. The game continues until one of the players is ruined. Find the probability that the second player is ruined, if:

Suppose that, under the conditions of the previous problem (see 7.32), player $A$ wins with probability $p$ and loses with probability $q$; in addition, a draw is possible with probability $r = 1 - p - q$. Find the probability that the second player is ruined.

Assuming that in the reproduction of bacteria by division (into two bacteria), the probability of a bacterium dividing during a time interval $\Delta$ is $\lambda \Delta + o(\Delta )$ as $\Delta \to 0$ and does not depend on the number of previous divisions or on the number of bacteria present, find the probability that, if there was one bacterium at time 0, there will be $i$ bacteria at time $t$.

A power line has $n$ mechanisms connected to it. The probability that a mechanism drawing power at time $t$ will stop drawing power before time $t+\Delta$ is $\alpha \Delta + o(\Delta )$ as $\Delta \to 0$. If at time $t$ a mechanism is not drawing power, the probability that it will start drawing power before time $t+\Delta$ is $\beta \Delta + o(\Delta )$. Set up the differential equations satisfied by the probabilities $P_r(t)$ that $r$ mechanisms are drawing power at time $t$.

Three reconnaissance aircraft are sent into enemy territory to determine the precise coordinates of an object to be shelled with missiles. $n$ missiles have been allocated for the destruction of the object. With precise coordinates for the object, the probability that a single missile destroys it is $p_1$; with imprecise coordinates, it is $p_2$. Each aircraft, before reaching the object's area, may be shot down by the enemy's air defenses with probability $p_3$. If an aircraft is not shot down, it radios the object's coordinates. Each such message is received by command headquarters with probability $p_4$. Receiving a message from a single aircraft suffices to determine the precise coordinates. Find the probability that the object is destroyed, taking the reconnaissance operation into account.

Among $n$ marksmen, four groups can be distinguished: $a_1$ excellent marksmen, $a_2$ good ones, $a_3$ mediocre ones, and $a_4$ poor ones. The probability of hitting the target with one shot for a marksman in the $i$-th group is $p_i$. Two marksmen are called at random and fire at the same target. Find the probability of at least one hit on the target.

A bus carries $n$ passengers. At the next stop, each of them gets off with probability $p$ independently of the others; furthermore, with probability $p_0$ no new passenger boards the bus, and with probability $1-p_0$ one new passenger boards. Find the probability that when the bus sets off again after the next stop, it will again carry $n$ passengers.

An urn contains two balls: one white and one black. Successive trials are performed, with the ball drawn returned to the urn each time. The number of trials is unlimited. What is the probability of eventually drawing: