Respuesta :
Given:
Numbers picked from 1 to 30 = 6 numbers
Number of ways = 593775
1 combination = $50,000
Given that the other 593774 combinations result in losing $1, let's find the expected value of losing the game.
Let's first find the number of ways of picking 6 numbers out of 30 numbers.
Apply the combination formula:
[tex]\begin{gathered} ^{30}C_6=\frac{30!}{6!(30-6)!} \\ \\ =\frac{30!}{6!(24)!} \\ \\ =\frac{30*29*28*27*26*25*24!}{6*5*4*3*2*1*24!} \\ \\ =\frac{427518000}{720} \\ \\ =593775\text{ ways} \end{gathered}[/tex]The probability of winning will be:
[tex]P(win\text{ 50000\rparen = }\frac{1}{593775}[/tex]The probability of losing is:
[tex]P(lose\text{ 1\rparen = }\frac{593774}{593775}[/tex]Hence, the expected value will be:
[tex]\begin{gathered} E=50000*\frac{1}{593775}+((-1)*\frac{593774}{593775}) \\ \\ E=50000*\frac{1}{593775}-\frac{593774}{593775} \\ \\ E=\frac{50000}{593775}-\frac{593774}{593775} \end{gathered}[/tex]Solving further, we have:
[tex]\begin{gathered} E=\frac{50000-593774}{593775} \\ \\ E=-\frac{543774}{593775} \\ \\ E=-0.916 \end{gathered}[/tex]Therefore, the expected value of this game is -0.916
ANSWER:
-0.916
