Question

Answer 1

Let us first consider the white balls.

5 balls out of a total of N=69 white balls picked by draw is of type winning balls. So white balls can be groups into 2 categories, winning balls (indicated by M=5), Other balls (indicated by N-M=69-5=64) .

Next a player picks 5 balls from 69 white balls. Let K=5 indicate the number of white balls picked by a player.

Let Y indicate the number of winning balls (matching numbers with the winning balls) in the 5 white balls that the player has picked.

Number of ways in which Y=y matching white balls (out of M=5 winning balls) and K-y=5-y white balls from 64 other balls can be picked by a player is

$\binom{M}{y}\binom{N-M}{k-y}=\binom{5}{y}\times \binom{64}{5-y}$

The total number of ways in which 5 balls can be selected from 69 is

65

the probability that Y=y white balls match the drawing is given by

Ply - y white baG

The above is a Hyper geometric distribution.

Now we consider the red balls.

There are 26 red balls, one of which is the winning ball.

The probability that a player selects the red ball which matches the draw is P(Red) = 1/26

The probability that a player selects the red ball which does not match the draw is P(no Red match) = (1-P(Red) = 25/26

Since the selection of white balls is independent of the red, the probability of y white balls+Red matching is

Ply white matches and red match-Ply-y white balls match) × P(Red match) 5 64 y) (5-y のー×26

and the probability of y white balls and no red balls matching is

Ply white balls match and no red ball match) = P(Y = y white balls match) × (1-Pl Red match)) ()) 25 64 5-25

Now we can fill the table to get the probability distribution of X, where X is the Money gained or lost (which is prize - ticket price of $2)

Prepare the following sheet (or use a calculator to calculate these)

1 Matching Numbers Prize Amount gained or lost (X) Probability 2 1. 5 white balls+ red ball -COMBIN(5,5) COMBIN(64,5-5)/COMBIN(69,5)*1/26 -COMBIN(5,5) COMBIN(64,5-5)/COMBIN (69,5)*25/26 -B2-2 3 2. 5 white balls -B3-2 1000000 -B4-2 COMBIN(5,4) COMBIN(64,5-4)/COMBIN(69,5)*1/26 -COMBIN(5,4)* COMBIN(64,5-4)/COMBIN(69,5)*25/26 4 3. 4 white balls+ red ball 50000 5 4.4 white balls 100 -B5-2 6 5. 3 white balls+ red ball -COMBIN(5,3) COMBIN(64,5-3)/COMBIN(69,5)*1/26 -COMBIN(5,3)*COMBIN(64,5-3)/COMBIN(69,5)*25/26 -COMBIN(5,2) COMBIN (64,5-2)/COMBIN(69,5)*1/26 -B6-2 100 7 6. 3 white balls 7 -B7-2 8 7. 2 white balls+ red ball -B8-2 9 8. 1 white balls+ red ball -COMBIN(5,1) COMBIN(64,5-1VCOMBIN(69,5) 1/26 4 -B9-2 -COMBIN(5,0) COMBIN(64,5-0)/COMBIN(69,5)*1/26 10 9. 0 white balls+ red ball -B10-2 4 11 Others -1-SUM(D2:D10) 0 -B11-2

get these

Amount gained or 1 Matching Numbers lost (X) Probability Prize 2 1.5 white balls+ red ball $100,000,000 $99,999,998 0.000000003 $1,000,000 $999,998 0.000000086 $49,9980.000001095 3 2. 5 white balls $50,000 4 3. 4 white balls+ red ball $100 S980.000027378 5 4.4 white balls $100 S980.000068994 6 5. 3 white balls+ red ball $7 $50.001724838 7 6. 3 white balls $7 S50.001425866 8 7.2 white balls+ red ball $4 20.010872229 9 8. 1 white balls+ red ball $4 20.026093351 10 9. 0 white balls+ red ball 11 Others $0 ($2) 0.959786160

The expected value of X is

$\begin{align*} E(X)&=\sum x P(X)\\ &=99999998\times 0.000000003+ 999998\times 0.000000086+ 49998\times 0.000001095+\\ &98\times 0.000027378+ 98\times 0.000068994+ 5\times 0.001724838+ 5\times 0.001425866+\\ &2\times 0.010872229+ 2\times 0.026093351+ -2\times 0.95978616\\ &=-\$1.34 \end{align*}$

The corresponding excel sheet is

X*Probability 1 Matching Numbers Prize Amount gained or lost(X) | Probability 2 1.5 white balls+ red ball -COMBIN(5,5)*COMBIN(64,5-5)/COMBIN(69,5) 1/26 C2*D2 -COMBIN(5,5) COMBIN(64,5-5)/COMBIN(69,5)* 25/26 C3 D3 COMBIN(5,4)*COMBIN(64,5-4YCOMBIN(69,5)*1/26 C4*D4 -COMBIN(5,4)*COMBIN(64,5-4)/COMBIN(69,5) 25/26 -C5*D5 -COMBIN(5,3) COMBIN(64,5-3)/COMBIN(69,5)*1/26 C6*D6 -B2-2 -B3-2 3 2.5 white balls 1000000 4 3. 4 white balls+ red ball B4-2 50000 100 5 4.4 white balls -B5-2 -B6-2 6 5.3 white balls+ red ball 100 7 6. 3 white balls COMBIN(S,3)*COMBIN(64,5-3 COMBIN(69,5)*25/26 -C7*D7 -COMBIN(5,2) COMBIN(64,5-2/COMBIN(69,5)"1/26 C8 D8 -COMBIN(5,1)*COMBIN(64,5-1)/COMBIN(69,5)*1/26 C9*D9 -COMBIN(5,0) COMBIN(64,5-O)/COMBIN(69,5)*1/26 C10 D10 B7-2 8 7.2 white balls+ red ball -B8-2 9 8.1 white balls+ red ball -B9-2 10 19. 0 white balls+ red ball B10-2 11 Others 1-SUM(D2:D10) 0 B11-2 C11 D11 12 Expected value SUM(E2:E11)

get this

Amount gained or lost (X) Probability X*Probability 1 Matching Numbers Prize $0.34 $0.09 2 1.5 white balls+ red ball $100,000,000 $99,999,998 0.000000003 $1,000,000 $999,998 0.000000086 $49,9980.000001095 3 2. 5 white balls $50,000 4 3. 4 white balls+ red ball $0.05 $100 S980.000027378 5 4.4 white balls $0.00 $100 S980.000068994 6 5. 3 white balls+ red ball $0.01 $7 S50.001724838 7 6. 3 white balls $0.01 $7 S50.001425866 8 7.2 white balls+ red ball $0.01 $4 20.010872229 $0.02 9 8. 1 white balls+ red ball $4 20.026093351 10 9. 0 white balls+ red ball $0.05 11 Others ($2) 0.959786160 ($1.92) ($1.34) $0 12 Expected value

ans: The expected value of the random variable X corresponding money gained or loss from buying the ticket and collecting a award is -$1.34