exercise:8f593ff6b0: Difference between revisions
From Stochiki
(Created page with "Suppose you are given the following: #<math>A_1, A_2, \ldots,A_n</math> are independent events defined on a sample space <math>\Omega</math> # <math>0 < P(A_j) < 1</math> for all <math>j</math> Which of the following statements is always true? <ul class="mw-excansopts"> <li><math>\sum_{i=1}^n P(A_i) < P(A_1 \cup \cdots \cup A_n)</math></li> <li><math>\sum_{i=1}^n P(A_i) = P(A_1 \cup \cdots \cup A_n)</math></li> <li><math>\sum_{i=1}^n P(A_i) < 1</math></li> <li...") |
No edit summary |
||
| (2 intermediate revisions by the same user not shown) | |||
| Line 7: | Line 7: | ||
<ul class="mw-excansopts"> | <ul class="mw-excansopts"> | ||
<li><math>\sum_{i=1}^n P(A_i) = P(A_1 \cup \cdots \cup A_n)</math></li> | <li><math>\sum_{i=1}^n P(A_i) = P(A_1 \cup \cdots \cup A_n)</math></li> | ||
<li><math>\sum_{i=1}^n P(A_i) < 1</math></li> | <li><math>\sum_{i=1}^n P(A_i) < 1</math></li> | ||
<li><math>\Omega</math> must have at least <math>2^n</math> points.</li> | <li><math>\Omega</math> must have at least <math>2^n</math> points.</li> | ||
<li><math>\Omega</math> must have at least <math>2n</math> points.</li> | <li><math>\Omega</math> must have at least <math>2n</math> points.</li> | ||
<li><math>\sum_{i=1}^j P(A_1 \cup \ldots \cup A_j) = P(A_1 \cup \ldots \cup A_n)</math> for <math>j < n </math></li> | |||
</ul> | </ul> | ||
'''References''' | |||
{{cite web |url=https://math.dartmouth.edu/~prob/prob/prob.pdf |title=Grinstead and Snell’s Introduction to Probability |last=Doyle |first=Peter G.|date=2006 |access-date=June 6, 2024}} | |||
Latest revision as of 14:15, 21 June 2024
Suppose you are given the following:
- [math]A_1, A_2, \ldots,A_n[/math] are independent events defined on a sample space [math]\Omega[/math]
- [math]0 \lt P(A_j) \lt 1[/math] for all [math]j[/math]
Which of the following statements is always true?
- [math]\sum_{i=1}^n P(A_i) = P(A_1 \cup \cdots \cup A_n)[/math]
- [math]\sum_{i=1}^n P(A_i) \lt 1[/math]
- [math]\Omega[/math] must have at least [math]2^n[/math] points.
- [math]\Omega[/math] must have at least [math]2n[/math] points.
- [math]\sum_{i=1}^j P(A_1 \cup \ldots \cup A_j) = P(A_1 \cup \ldots \cup A_n)[/math] for [math]j \lt n [/math]
References
Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.