418531 ภาคต้น 2552/โจทย์ปัญหาความน่าจะเป็น I/เฉลยข้อ 8

เราจะพิสูจน์ข้อความในโจทย์ด้วย induction บน

(Base Case) มีค่าเท่ากับ 1 เราได้ว่า

Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 1071395950
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Wed, 25 Feb 2026 21:23:26 GMT


Sensitive client information

IP address: 158.108.32.49

$\Pr(E_{1})\geq \Pr(E_{1})-(1-1)$

(Induction Case) สมมติให้ข้อความในโจทย์เป็นจริง ให้ เป็นเหตุการณ์ใดๆ

เราได้ว่า

Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 1073163192 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Wed, 25 Feb 2026 21:23:22 GMT Sensitive client information IP address: 158.108.32.49 $\Pr(E_{1}\cap E_{2}\cap \dotsb \cap E_{n}\cap E_{n+1})\,$	$=\,$
		Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 1071820836 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Wed, 25 Feb 2026 21:23:22 GMT Sensitive client information IP address: 158.108.32.49 $1-\Pr({\overline {E_{1}}}\cup {\overline {E_{2}}}\cup \dotsb \cup {\overline {E_{n}}}\cup {\overline {E_{n+1}}})\,$
			เหตุผล: โดย union bound เราได้ว่า $\Pr({\overline {E_{1}}}\cup {\overline {E_{2}}}\cup \dotsb \cup {\overline {E_{n}}}\cup {\overline {E_{n+1}}})\leq \Pr({\overline {E_{1}}}\cup {\overline {E_{2}}}\cup \dotsb \cup {\overline {E_{n}}})+\Pr({\overline {E_{n+1}}})\,$
		Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 1069136173 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Wed, 25 Feb 2026 21:23:23 GMT Sensitive client information IP address: 158.108.32.49 $1-\Pr({\overline {E_{1}\cap E_{2}\cap \dotsb \cap E_{n}}})-1+\Pr(E_{n+1})\,$
		Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 1070448169 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Wed, 25 Feb 2026 21:23:23 GMT Sensitive client information IP address: 158.108.32.49 $\Pr(E_{1}\cap E_{2}\cap \dotsb \cap E_{n})-1+\Pr(E_{n+1})\,$
		Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 10327353 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Wed, 25 Feb 2026 21:23:24 GMT Sensitive client information IP address: 158.108.32.49 $\Pr(E_{1})+\Pr(E_{2})+\dotsb +\Pr(E_{n})-(n-1)-1+\Pr(E_{n+1})\,$	ด้วยสมมติฐานของ induction

ฉะนั้นเราสามารถสรุปได้ว่าข้อความในโจทย์เป็นจริงสำหรับจำนวนเต็มบวก ทุกจำนวน