Python 2, 103 93 89 bytes

I combined my earlier answer with Chas Browns's answer and got something a few bytes shorter.

The random number will be between 0 and 2**31-1 inclusive.

from random import*

i=input()

while'+'in i:1

print["Hello World",randrange(2**31)][i<'!']

Try it online!

Previous versions:

103 bytes

from random import*

exec['print"Hello World"','while 1:1','print randint(0,2**31-1)'][cmp(input(),'*')]

93 bytes

from random import*

i=cmp(input(),'*')

while i>0:1

print["Hello World",randint(0,2**31-1)][i]

Jelly, 21 bytes

Ø%HX’



“½,⁾ẇṭ»

OS¹Ḃ¿£

A full program.

Try it online!

How?

Ø%HX’ - Link 1, get_random(): no argument

Ø%    - literal 2^32

  H   - half = 2^31

   X  - random integer in [1,n=2^31]

    ’ - decrement



 - Link 2, get_zero(): no argument

 - no instruction so yield the implicit input of zero



“½,⁾ẇṭ» - Link 3, get_hello_world(): no argument

“½,⁾ẇṭ» - list of characters from dictionary words "Hello"+" World"



OS¹Ḃ¿£ - Main Link: list of characters          e.g.: "*"  or   " * "     or  "*+*"

O      - to ordinals                                  [42]      [32,42,32]    [42,43,42]

 S     - sum                                          42        106           127

    ¿  - while...

   Ḃ   - ...condition: modulo 2 (i.e. is odd?)        0         0             1

  ¹    - ...do: identity (get value of own input)                             127

     £ - call link at (modular) index                 42        106

       - -- since 3 non-main Links these are          3         1

JavaScript (ES7), 66 bytes

s=>s[1]?s<'!'?Math.random()*2**31|0:eval(`for(;;);`):'Hello World'

Try it online! (Hello World)

Try it online! (random number)

Try it online! (infinite loop)

Japt, 22 bytes

Throws an overflow error upon entering the infinite loop of the third programme but, theoretically, with enough memory (which we may assume for the purposes of code-golf), would run forever.

Å?¢?ß:2pHÉ ö:`HÁM Wld

Try programme 1
Try programme 2
Try programme 3

Python 2, 91 89 bytes

from random import*

def f(p):

 while'*'<p:p

 print['Hello World',randrange(2**31)][p<'!']

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2 bytes thanks to Jonathan Allan.

R, 69 bytes

switch(scan(,""),"*"="Hello, World!"," * "=sample(2^31,1)-1,repeat{})

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switch tries to match the named arguments and if there's no match, selects the first unnamed one after the first, which in this case is the infinite loop repeat{}.

C (gcc), 66 63 bytes

Thanks to attinat for the -3 bytes.

I only have to check the second character: if the LSB is set, it's a + (thus the program is "*+*") and the program loops. After that, if it's a NUL, the program was "*" and we display Hello World; otherwise, it displays a random value ("*", the only other option left.)

f(char*s){for(s++;*s&1;);printf(*s?"%d":"Hello World",rand());}

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Keg, 30 26 bytes

?!1=[_Hello World| =[__~|{

Essentially, takes the input program and:

• Checks to see if the input length is 1, printing "Hello World" if true


• Checks to see if the last character is a space, and prints a random number


• Otherwise runs an infinite loop

Then implicitly print the stack.

?                               #Get input from the user

 !1=                            #Compare the stack's length to 1

    [_Hello World           #Push "Hello, World!" to the stack

                     | =        #See if top item is a space

                        [__~|{  #If so, generate a random number, otherwise, infinite loop.

4 bytes saved due to the fact that hello world doesnt need punctuation.

Try it online! Old version

Try it online! New version

Brachylog, 26 bytes

l₃∈&hṢ∧2^₃₂-₁ṙw∨Ḥk⟨h₅ct₆⟩w

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Takes the program as a string through the input variable, and ignores the output variable. Heavily exploits the guarantee that the input is only ever one of the three valid programs: any length-three input will behave like either " * " or "*+*" depending on whether or not the first character is a space, and any other input will behave like "*".

l₃                            The input has length 3

  ∈                           and is an element of something,

   &h                         and the input's first element

     Ṣ                        is a space

  ∈                           (if not, try some other thing it's an element of),

      ∧2^₃₂-₁                 so take 2,147,483,647 and

             ṙw               print a random number between 0 and it inclusive.

               ∨              If the input's length isn't 3,

                  ⟨h₅ct₆⟩w    print the first 5 and last 6 characters of

                Ḥk            "Hello, World!" without its last character.

Perl 5 -p, 43 39 bytes

$_=/ /?0|rand~0:/+/?redo:"Hello World"

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C# (Visual C# Interactive Compiler), 71 bytes

s=>{for(;s=="*+*";);return"*"==s?"Hello World":new Random().Next()+"";}

Try it online!

Ruby -n, 47 bytes

puts~/ /?rand(1<<31):~/+/?loop{}:"Hello World"

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Wolfram Language (Mathematica), 65 bytes

#/.{"*"->"Hello World"," * "->RandomInteger[2^31-1],_:>0~Do~∞}&

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Charcoal, 30 bytes

Ｗ№θ*Ｆ⁼θ*≔Hello Worldθ∨θＩ‽Ｘ²¦³¹

Try it online! Link is to verbose version of code. Abuses Charcoal's default input format which splits on spaces if there is only one line, thus the random number input actually looks like three inputs. Explanation:

Ｗ№θ*

Repeat while the first input contains a *.

Ｆ⁼θ*

If the first input is a * only...

≔Hello Worldθ

... then replace it with Hello World, thus causing the loop to terminate. *+* doesn't get replaced, resulting in an infinite loop.

∨θ

If the first input is not empty then output it.

Ｉ‽Ｘ²¦³¹

But if it is empty then output a random integer in the desired range.

Befunge-93, 50 bytes

~"*"-_~1+#^_"dlroW olleH">:#,_@.%*2**:*::*88:?1#+<

Try it online!

Late right now. Will likely improve and/or add an explanation tomorrow.

Jelly, 23 22 bytes

OS¹Ḃ¿%3ịØ%HX’,“½,⁾ẇṭ»¤

Try it online!

A monadic link taking a single argument and returning Hello World, a random 31 bit integer or looping infinitely as per the spec.

All options: * * *+*

Explanation

O                      | Convert to codepoints

 S                     | Sum

  ¹Ḃ¿                  | Loop the identity function while odd 

     %3                | Mod 3

       ị             ¤ | Index into the following as a nilad:

        Ø%             | - 2 ** 32

          H            | - Halved

           X           | - Random integer in the range 1..2**31

            ’          | - Decrease by 1 

             ,“½,⁾ẇṭ»  | - Pair with "Hello World"