Probabilidade: Exercicios Resolvidos
Favorable pairs: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 outcomes. [ P(\textsum = 7) = \frac636 = \frac16 \approx 0.1667 ]
Given: [ P(D) = 0.001,\quad P(T^+|D) = 0.99,\quad P(T^+|\neg D) = 0.01 ] By Bayes' theorem: [ P(D|T^+) = \fracD)P(D)\neg D)P(\neg D) ] [ = \frac0.99 \times 0.0010.99 \times 0.001 + 0.01 \times 0.999 ] [ = \frac0.000990.00099 + 0.00999 = \frac0.000990.01098 \approx 0.09016 ]
After removing 1 red, left: 3 red + 6 blue = 9 marbles. [ P(B_2 | R_1) = \frac69 = \frac23 ] probabilidade exercicios resolvidos
About 9.02%. Despite high accuracy, low prevalence means most positives are false positives. Exercise 5: Binomial Probability Problem: A fair coin is tossed 5 times. What is the probability of getting exactly 3 heads? Solution: Binomial with ( n=5, k=3, p=0.5 ): [ P(X=3) = \binom53 (0.5)^3 (0.5)^2 = 10 \times (0.5)^5 ] [ = 10 \times \frac132 = \frac1032 = \frac516 = 0.3125 ]
( \frac516 ). Exercise 6: Complement and "At Least One" Problem: If you roll a fair die 3 times, what is the probability of getting at least one 6? Solution: Easier to use complement: [ P(\textat least one 6) = 1 - P(\textno 6) ] Probability of no 6 in one roll = ( \frac56 ). [ P(\textno 6 in 3 rolls) = \left(\frac56\right)^3 = \frac125216 ] [ P(\textat least one 6) = 1 - \frac125216 = \frac91216 \approx 0.4213 ] Favorable pairs: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
Red suits = hearts + diamonds → 2 suits × 3 face cards = 6 [ P = \frac652 = \frac326 \approx 0.1154 ]
3 face cards per suit × 4 suits = 12 face cards [ P = \frac1252 = \frac313 \approx 0.2308 ] Despite high accuracy, low prevalence means most positives
( \frac91216 ). Summary of Key Formulas Used | Concept | Formula | |---------|---------| | Classical probability | ( P(A) = \frac ) | | Conditional probability | ( P(A|B) = \fracP(A \cap B)P(B) ) | | Multiplication rule | ( P(A \cap B) = P(A)P(B|A) ) | | Bayes' theorem | ( P(A|B) = \fracA)P(A)P(B) ) | | Binomial probability | ( P(X=k) = \binomnk p^k (1-p)^n-k ) | | Complement rule | ( P(A) = 1 - P(\neg A) ) | These exercises illustrate the most common reasoning patterns in introductory probability. Practice with variations (more dice, different decks, multiple events) to build intuition.