3.30. In Laplace's rule of succession (Example 5e ), suppose that the first n flips resulted in r heads and n -r tails. Show that the probability that the (n+1) flip turns up heads is (r + 1)/(n + 2). To do so, you will have to prove and use the identity parts yields m [*(1-x) - dy = Hint: To prove the identity, let 0.04 n!m! (n + m + 1)! m let c(n.m)-[(1-3) "dy. Integrating by m c(n,m) = 7₁ C(n + 1, m − 1) n+ 1 Starting with C(n, 0) = 1/(n + 1), prove the identity by induction on m. natinal but philosophically minded friend of
3.30. In Laplace's rule of succession (Example 5e ), suppose that the first n flips resulted in r heads and n -r tails. Show that the probability that the (n+1) flip turns up heads is (r + 1)/(n + 2). To do so, you will have to prove and use the identity parts yields m [*(1-x) - dy = Hint: To prove the identity, let 0.04 n!m! (n + m + 1)! m let c(n.m)-[(1-3) "dy. Integrating by m c(n,m) = 7₁ C(n + 1, m − 1) n+ 1 Starting with C(n, 0) = 1/(n + 1), prove the identity by induction on m. natinal but philosophically minded friend of
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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3.30. How can I show that?
![**3.30. In Laplace’s rule of succession (Example 5e), suppose that the first n flips resulted in r heads and n - r tails. Show that the probability that the (n + 1) flip turns up heads is (r + 1)/(n + 2). To do so, you will have to prove and use the identity**
\[
\int_0^1 y^n (1-y)^m \, dy = \frac{n!m!}{(n+m+1)!}
\]
**Hint:** To prove the identity, let
\[
C(n, m) = \int_0^1 y^n (1-y)^m \, dy
\]
Integrating by parts yields
\[
C(n, m) = \frac{m}{n+1} C(n+1, m-1)
\]
Starting with \(C(n, 0) = 1/(n+1)\), prove the identity by induction on m.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0618968-6b5a-406b-a92d-661b061825b7%2F07296499-004a-4181-9814-c7179ace4a9d%2Fs5c2ul5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**3.30. In Laplace’s rule of succession (Example 5e), suppose that the first n flips resulted in r heads and n - r tails. Show that the probability that the (n + 1) flip turns up heads is (r + 1)/(n + 2). To do so, you will have to prove and use the identity**
\[
\int_0^1 y^n (1-y)^m \, dy = \frac{n!m!}{(n+m+1)!}
\]
**Hint:** To prove the identity, let
\[
C(n, m) = \int_0^1 y^n (1-y)^m \, dy
\]
Integrating by parts yields
\[
C(n, m) = \frac{m}{n+1} C(n+1, m-1)
\]
Starting with \(C(n, 0) = 1/(n+1)\), prove the identity by induction on m.
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