Calculating the Probability of Drawing Exactly 2 White Balls in a Binomial Distribution

Extract

[...] So X can take the values: X = 3 Given that to determine the probability law, one must find the probabilities obtain 4 white balls event E : obtain 0 white balls: event B : obtain 1 white balls: event C : obtain 2 white balls: event A : obtain 3 white balls: event D The event C decomposes into two incompatible sub-events 2 to 2 : one white ball with one black ball of and 0 white balls with 2 black balls of : we draw 0 white balls with 2 black balls of and 1 white ball with 1 black ball of are incompatible is the set of all possibilities to choose 1 white ball among 3 with 1 black ball among 2 of and 0 white ball among 2 with 2 black balls among 3 of is the set of all possibilities to choose 0 white balls among 3 with 2 black balls among 2 of and 1 white ball among 2 with 1 black ball among 3 of Event D decomposes into two incompatible sub-events 2 by : we draw 2 white balls with 0 black balls of and 1 white ball with 1 black ball of : we draw 1 white ball with 1 black ball of and 2 white balls with 2 black balls of and being incompatible we have : is the set of all possibilities to choose 2 white balls out of 3 with 0 black balls out of 2 and 1 white ball out of 2 with 1 black ball out of 3 of is the set of all possibilities to choose 1 white ball among 3 with 1 black ball among 2 of and 2 white balls among 2 with 0 black balls among 3 of one draws 2 white balls out of 3 with 0 black balls out of 2 of and 2 white balls out of 2 with 0 black balls out of 3 of From where the following probability law comes: 1 2 3 4 0,03 0,24 0,46 0,24 0,03 Class Review 1here S is the mathematical expectation On The player can expect to hit on average 2 F and the bet is 2.50 The mathematical expectation is lower than the bet. The game is not favorable for the player. We want to calculate the probability of the event assuming A occurs A A Reminder on Bernoulli's equation 1 S A Bernoulli trial is a random experiment with exactly 2 possible outcomes. [...]

[...] Binomial distribution schema: succession of identical and independent Bernoulli trials Binomial law With p : probabilities of success With q : probabilities of failure The fact of drawing exactly 2 balls is called a success having exactly 2 white balls is a Bernoulli trial with exactly two outcomes. success draw 2 white balls failure not to draw 2 white balls We repeat this test 10 times, systematically putting the drawn balls back in the urn. We therefore have a succession of identical and independent Bernoulli trials. [...]

[...] Probabilities - Baccalaureate S Subject: Group II bis (groups II-III), June 1996 Let us calculate the set of all possible outcomes of this random experiment, i.e., the number of elements of the set ? the universe be card(?) We draw simultaneously 2 balls from urn 1 and 2 balls from urn 2 The draw in the urn is simultaneous. There is no order or possible repetitions. [...]

[...] Let A be the event among the 4 balls drawn where we get exactly 2 white balls. This event can be decomposed into 3 incompatible events 2 by 2. : We drew 2 white balls with 0 black balls of and 0 white ball with 2 black balls of : We drew 1 white balls with 1 black ball of and 1 white ball with 1 black balls of : We drew 0 white balls with 2 black balls of and 2 white balls with 0 black balls of the events are incompatible in pairs Card is the set of all possibilities aiming to choose 2 balls out of 3 with 0 balls out of 2 and choose 0 ball out of 2 with 2 balls out of 3 of Card is the set of all possibilities aiming to choose 1 ball out of 3 with 1 ball out of 2 of" and choose 1 ball out of 2 with 1 ball out of 3 of Card Return to calculate the entire set of possibilities aiming to choose 0 ball among 3 with 2 balls among 2 of and choose 2 balls among 2 with 0 ball among 3 of X denotes the random variable that associates with each draw the number of white balls obtained. [...]

[...] We have a Bernoulli scheme with parameter n = 10 and probability of success. Let's calculate Let G be the event of getting exactly 2 white balls in the draw Card the number of possibilities to choose 2 balls out of 3 Card le nombre de possibilité de choisir 2 boules parmi 5 The random variable X which counts the number of times we obtained exactly 2 balls follows a binomial distribution B 0.3) Here we are looking for so that the opposite event occurs exactly 0 times 2 white balls in the 10 draws, that is Secondary question: Calculate the probability of getting exactly 4 times two white balls. [...]