CHAPTER 4 Special Probability Distributions
Read QR In NoneUsing Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications.
Encoding QR Code 2d Barcode In NoneUsing Barcode encoder for Software Control to generate, create QR Code image in Software applications.
Method 2 (using formula) (a) P(3 heads) 3 1 3 1 0 a ba b a b 2 3 2 1 8 3 8
QR-Code Decoder In NoneUsing Barcode scanner for Software Control to read, scan read, scan image in Software applications.
QR-Code Printer In C#.NETUsing Barcode generator for Visual Studio .NET Control to generate, create QR Code image in VS .NET applications.
(b) P(2 tails and 1 head) (c) P(at least 1 head)
QR Code ISO/IEC18004 Printer In VS .NETUsing Barcode encoder for ASP.NET Control to generate, create QR Code 2d barcode image in ASP.NET applications.
QR Code Maker In Visual Studio .NETUsing Barcode creation for .NET framework Control to generate, create QR Code image in Visual Studio .NET applications.
3 1 2 1 1 a ba b a b 2 2 2 P(1, 2, or 3 heads) P(1 head)
QR Code ISO/IEC18004 Maker In Visual Basic .NETUsing Barcode maker for VS .NET Control to generate, create Quick Response Code image in Visual Studio .NET applications.
UPC - 13 Creator In NoneUsing Barcode creation for Software Control to generate, create EAN13 image in Software applications.
P(2 heads)
Generate USS Code 39 In NoneUsing Barcode drawer for Software Control to generate, create Code 39 image in Software applications.
Make Data Matrix ECC200 In NoneUsing Barcode creator for Software Control to generate, create ECC200 image in Software applications.
P(3 heads) 3 1 3 1 0 a ba b a b 2 3 2 7 8 ,
Create Bar Code In NoneUsing Barcode drawer for Software Control to generate, create bar code image in Software applications.
EAN / UCC - 13 Maker In NoneUsing Barcode drawer for Software Control to generate, create EAN / UCC - 14 image in Software applications.
3 1 1 1 2 a ba b a b 2 1 2 Alternatively, P(at least 1 head)
Postnet Generator In NoneUsing Barcode generation for Software Control to generate, create Postnet image in Software applications.
Generate Bar Code In Visual Basic .NETUsing Barcode printer for .NET Control to generate, create bar code image in VS .NET applications.
3 1 2 1 1 a ba b a b 2 2 2
Generate GS1 RSS In JavaUsing Barcode printer for Java Control to generate, create GS1 DataBar Limited image in Java applications.
Universal Product Code Version A Decoder In JavaUsing Barcode decoder for Java Control to read, scan read, scan image in Java applications.
P(no head) 3 1 0 1 3 a ba b a b 2 0 2 P(0 tails or 1 tail) P(0 tails) P(1 tail) 3 1 2 1 a ba b a b 2 2 2 1 2 7 8
Recognizing UPC - 13 In JavaUsing Barcode reader for Java Control to read, scan read, scan image in Java applications.
European Article Number 13 Reader In NoneUsing Barcode decoder for Software Control to read, scan read, scan image in Software applications.
(d) P(not more than 1 tail)
Paint Data Matrix 2d Barcode In NoneUsing Barcode printer for Font Control to generate, create Data Matrix image in Font applications.
EAN13 Generator In NoneUsing Barcode printer for Online Control to generate, create EAN / UCC - 13 image in Online applications.
3 1 3 1 0 a ba b a b 2 3 2
It should be mentioned that the notation of random variables can also be used. For example, if we let X be the random variable denoting the number of heads in 3 tosses, (c) can be written P(at least 1 head) P(X 1) P(X 1) P(X 2) P(X 3) 7 8
We shall use both approaches interchangeably.
4.2. Find the probability that in five tosses of a fair die, a 3 will appear (a) twice, (b) at most once, (c) at least two times.
Let the random variable X be the number of times a 3 appears in five tosses of a fair die. We have Probability of 3 in a single toss Probability of no 3 in a single toss (a) P(3 occurs twice) P(X 2) P(X 5 1 2 5 3 a ba b a b 6 2 6 1) P(X 0) 625 3888 P(X 1) p q 1 6 1 p 5 6
(b) P(3 occurs at most once)
5 1 0 5 5 a ba b a b 6 0 6 3125 7776 (c) P(3 occurs at least 2 times) P(X P(X 2) 2) P(X 3) P(X 4) 3125 7776
5 1 1 5 4 a ba b a b 6 1 6 3125 3888
5) 5 1 5 5 0 a ba b a b 6 5 6
5 1 2 5 3 a ba b a b 6 2 6 625 3888 125 3888
5 1 3 5 2 a ba b a b 6 3 6 25 7776 1 7776
5 1 4 5 1 a ba b a b 6 4 6 763 3888
CHAPTER 4 Special Probability Distributions
4.3. Find the probability that in a family of 4 children there will be (a) at least 1 boy, (b) at least 1 boy and at least 1 girl. Assume that the probability of a male birth is 1> 2.
(a) P(1 boy) P(3 boys) Then P(at least 1 boy) P(1 boy) 1 4 Another method P(at least 1 boy) (b) P(at least 1 boy and at least 1 girl) 1 1 1 P(no boy) P(no boy) 1 16 1 16 1 1 4 a b 2 1 1 16 15 16 3 8 1 4 P(2 boys) 1 16 15 16 P(3 boys) P(4 boys) 4 1 1 1 3 a ba b a b 2 1 2 4 1 3 1 1 a ba b a b 2 3 2 1 , 4 1 , 4 P(2 boys) P(4 boys) 4 1 2 1 2 a ba b a b 2 2 2 4 1 4 1 0 a ba b a b 2 4 2 3 8 1 16
P(no girl) 7 8
We could also have solved this problem by letting X be a random variable denoting the number of boys in families with 4 children. Then, for example, (a) becomes P(X 1) P(X 1) P(X 2) P(X 3) P(X 4) 15 16
4.4. Out of 2000 families with 4 children each, how many would you expect to have (a) at least 1 boy, (b) 2 boys, (c) 1 or 2 girls, (d) no girls
Referring to Problem 4.3, we see that (a) Expected number of families with at least 1 boy (b) Expected number of families with 2 boys (c) P(1 or 2 girls) P(1 girl) P(1 boy) P(2 girls) P(2 boys) 1 4 3 8 5 8 1250 125 2000a 15 b 16 1875 3 2000a b 8 750
2000 P(2 boys)
