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S = {HH, HT, TH, TT} of equally likely simple events. We are interested in the following events: E₁ = a head on the first toss E₂ = a tail on the first toss E3= a tail on the second toss E4 = a head on the second toss For each pair of events, discuss whether they are independent and whether they are mutually exclusive. (A) E₁ and E4 (B) E₁ and E₂ 50. For each pair of events (see Problem 49), discuss whether they are independent and whether they are mutually exclusive. (A) E, and E3 (B) E3 and E4 51 In 2 throws of Pro drav ball: even 59. 60.

S = {HH, HT, TH, TT} of equally likely simple events. We are interested in the following events: E₁ = a head on the first toss E₂ = a tail on the first toss E3= a tail on the second toss E4 = a head on the second toss For each pair of events, discuss whether they are independent and whether they are mutually exclusive. (A) E₁ and E4 (B) E₁ and E₂ 50. For each pair of events (see Problem 49), discuss whether they are independent and whether they are mutually exclusive. (A) E, and E3 (B) E3 and E4 51 In 2 throws of Pro drav ball: even 59. 60.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please help with #50. Thank you so much.
### Probability and Events: Coin Tosses and Dice Rolls #### Problems: **47.** (A) \( P(M \cap S) \) (B) \( P(R) \) **48.** (A) \( P(N \cap R) \) (B) \( P(S) \) **49. Coin Toss Experiment:** - A fair coin is tossed twice. Consider the sample space \( S = \{ HH, HT, TH, TT \} \) of equally likely simple events. - We are interested in the following events: - \( E_1 \): a head on the first toss - \( E_2 \): a tail on the first toss - \( E_3 \): a tail on the second toss - \( E_4 \): a head on the second toss For each pair of events, discuss whether they are independent and whether they are mutually exclusive. **50. Event Analysis:** - For each pair of events (see Problem 49), discuss whether they are independent and whether they are mutually exclusive. - (A) \( E_1 \) and \( E_3 \) - (B) \( E_3 \) and \( E_4 \) **51. Dice Roll Probability:** - In two throws of a fair die, what is the probability that you will get an even number on each throw? An even number on the first or second throw? **52. Threshold Probability:** - In two throws of a fair die, what is the probability that you will get at least 5 on each throw? At least 5 on the first or second throw? **53. Card Drawing Probability:** - Two cards are drawn in succession from a standard 52-card deck. What is the probability that the first card is a club and the second card is a heart? - (A) If the cards are drawn without replacement? - (B) If the cards are drawn with replacement? **54. Red Cards Probability:** - Two cards are drawn in succession from a standard 52-card deck. What is the probability that both cards are red? This series of questions combines concepts of probability, independence, mutual exclusivity, and basic combinatorics to explore foundational principles in the field.
Transcribed Image Text:### Probability and Events: Coin Tosses and Dice Rolls #### Problems: **47.** (A) \( P(M \cap S) \) (B) \( P(R) \) **48.** (A) \( P(N \cap R) \) (B) \( P(S) \) **49. Coin Toss Experiment:** - A fair coin is tossed twice. Consider the sample space \( S = \{ HH, HT, TH, TT \} \) of equally likely simple events. - We are interested in the following events: - \( E_1 \): a head on the first toss - \( E_2 \): a tail on the first toss - \( E_3 \): a tail on the second toss - \( E_4 \): a head on the second toss For each pair of events, discuss whether they are independent and whether they are mutually exclusive. **50. Event Analysis:** - For each pair of events (see Problem 49), discuss whether they are independent and whether they are mutually exclusive. - (A) \( E_1 \) and \( E_3 \) - (B) \( E_3 \) and \( E_4 \) **51. Dice Roll Probability:** - In two throws of a fair die, what is the probability that you will get an even number on each throw? An even number on the first or second throw? **52. Threshold Probability:** - In two throws of a fair die, what is the probability that you will get at least 5 on each throw? At least 5 on the first or second throw? **53. Card Drawing Probability:** - Two cards are drawn in succession from a standard 52-card deck. What is the probability that the first card is a club and the second card is a heart? - (A) If the cards are drawn without replacement? - (B) If the cards are drawn with replacement? **54. Red Cards Probability:** - Two cards are drawn in succession from a standard 52-card deck. What is the probability that both cards are red? This series of questions combines concepts of probability, independence, mutual exclusivity, and basic combinatorics to explore foundational principles in the field.
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