By Vladimir I. Rotar
Easy NotionsSample house and EventsProbabilitiesCounting TechniquesIndependence and Conditional ProbabilityIndependenceConditioningThe Borel-Cantelli TheoremDiscrete Random VariablesRandom Variables and VectorsExpected ValueVariance and different Moments. Inequalities for DeviationsSome simple DistributionsConvergence of Random Variables. The legislation of enormous NumbersConditional ExpectationGenerating features. Branching Read more...
summary: uncomplicated NotionsSample area and EventsProbabilitiesCounting TechniquesIndependence and Conditional ProbabilityIndependenceConditioningThe Borel-Cantelli TheoremDiscrete Random VariablesRandom Variables and VectorsExpected ValueVariance and different Moments. Inequalities for DeviationsSome easy DistributionsConvergence of Random Variables. The legislations of huge NumbersConditional ExpectationGenerating features. Branching procedures. Random stroll RevisitedBranching tactics producing services Branching techniques Revisited extra on Random WalkMarkov ChainsDefinitions and Examples. chance Distributio
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Example text
E) Write a formula for the probability that at least one player will get a whole suit in terms of the above probabilities. ) 23. There are ten pairs of shoes in a closet. (a) If you choose two shoes at random, what is the probability that it will be a pair? (b) You have chosen four shoes at random. Find the probability that among them, there will be at least one pair in two ways: directly and computing the probability of the complement of the event in hand. 24. Let P(A1 ) = P(A2 ) = 1. Show that P(A1 ∪ A2 ) = P(A1 A2 ) = 1.
1 1 1 , 2 + n+1 ]. So, Δ1 = [0, 1], Let us fix a point, say, 12 , and set the interval Δn = [ 12 − n+1 1 5 1 3 Δ2 = [ 6 , 6 ], Δ3 = [ 4 , 4 ], and so on. One may say, that the intervals Δn are “squeezing” to the point 1/2 as n → ∞. More rigorously, if we denote by [1/2] the set that consists of only one point 1/2, then ∩∞ 1 Δn = [1/2]. Thus, since the sequence Δn is non-increasing, we have limn→∞ Δn = ∩∞ Δ = [1/2]. 1 n For simplicity, let us write P(Δ) for the probability that the point chosen is in Δ.
B) Now, let us compute the probability that at most 5 customers bought a dairy product. Proceeding in a straightforward fashion, we may add up all probabilities P(Bk,7 ) for k = 1. , 5, but this is tedious. Instead, we may compute the probability that the number of customers is larger than 5, that is, P(B6,7 ) + P(B7,7 ), which is shorter. After that we will subtract this probability from one. 75)0 6 7 7! 7! 0013. 1! 0! 9987. 2) and after simple calculations get P(no successes in n trials) = qn .