Hazard game

In the Sportka hazard game, 6 numbers out of 49 are drawn. What is the probability that we will win:

a) second prize (we guess 5 numbers correctly)
b) the third prize (we guess 4 numbers correctly)?

Correct result:

p₁ =  0.002 %
p₂ =  0.097 %

Solution:

$C_{{ 5}}(6) = \dbinom{ 6}{ 5} = \dfrac{ 6! }{ 5!(6-5)!} = \dfrac{ 6 } { 1 } = 6 \ \\ C_{{ 1}}(43) = \dbinom{ 43}{ 1} = \dfrac{ 43! }{ 1!(43-1)!} = \dfrac{ 43 } { 1 } = 43 \ \\ C_{{ 6}}(49) = \dbinom{ 49}{ 6} = \dfrac{ 49! }{ 6!(49-6)!} = \dfrac{ 49 \cdot 48 \cdot 47 \cdot 46 \cdot 45 \cdot 44 } { 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 } = 13983816 \ \\ n=49 \ \\ o=6 \ \\ s=n-o=49-6=43 \ \\ \ \\ p_{1}=100 \cdot \ \dfrac{ { { o } \choose 5 } \cdot \ { { s } \choose 1 } }{ { { n } \choose o } }=100 \cdot \ \dfrac{ 6 \cdot \ 43 }{ 13983816 }=0.002 \%$

$C_{{ 4}}(6) = \dbinom{ 6}{ 4} = \dfrac{ 6! }{ 4!(6-4)!} = \dfrac{ 6 \cdot 5 } { 2 \cdot 1 } = 15 \ \\ C_{{ 2}}(43) = \dbinom{ 43}{ 2} = \dfrac{ 43! }{ 2!(43-2)!} = \dfrac{ 43 \cdot 42 } { 2 \cdot 1 } = 903 \ \\ p_{2}=100 \cdot \ \dfrac{ { { o } \choose 4 } \cdot \ { { s } \choose 2 } }{ { { n } \choose o } }=100 \cdot \ \dfrac{ 15 \cdot \ 903 }{ 13983816 }=0.097 \%$

Try another example

Please write to us with your comment on the math problem or ask something. Thank you for helping each other - students, teachers, parents, and problem authors.

Showing 0 comments:

Next similar math problems:

• Two aces
  From a 32 card box we randomly pick 1 card and then 2 more cards. What is the probability that last two drawn cards are aces?
• One green
  In the container are 45 white and 15 balls. We randomly select 5 balls. What is the probability that it will be a maximum one green?
• travel agency
  Small travel agency offers 5 different tours at honeymoon. What is the probability that the bride and groom choose the same tour (they choose independently)?
• Cards
  The player gets 8 cards of 32. What is the probability that it gets a) all 4 aces b) at least 1 ace
• A book
  A book contains 524 pages. If it is known that a person will select any one page between the pages numbered 125 and 384, find the probability of choosing the page numbered 252 or 253.
• Internet anywhere
  In school, 60% of pupils have access to the internet at home. A group of 8 students is chosen at random. Find the probability that a) exactly 5 have access to the internet. b) At least 6 students have access to the internet
• Division
  Division has 18 members: 10 girls and 6 boys, 2 leaders. How many different patrols can be created, if one patrol is 2 boys, 3 girls and 1 leader?
• A student
  A student is to answer 8 out of 10 questions on the exam. a) find the number n of ways the student can choose 8 out of 10 questions b) find n if the student must answer the first three questions c) How many if he must answer at least 4 of the first 5 ques
• Calculation of CN
  Calculate: ?
• Menu
  On the menu are 12 kinds of meal. How many ways can we choose four different meals into the daily menu?
• Trinity
  How many different triads can be selected from the group 43 students?
• Fish tank
  A fish tank at a pet store has 8 zebra fish. In how many different ways can George choose 2 zebra fish to buy?
• The box
  The box contains five chocolate, three fruit, and two menthol candies. We choose sweets at random from the box. What is the probability that we will take out one chocolate, one fruit, and one menthol candy without a return?
• Green - Red
  We have 5 bags. Each consist one green and 2 red balls. From each we pull just one ball. What is the probability that we doesn't pull any green ball?
• Balls
  From the urn in which are 7 white balls and 17 red, gradually drag 3-times without replacement. What is the probability that pulls balls are in order: red red red?
• Hearts
  5 cards are chosen from a standard deck of 52 playing cards (13 hearts) with replacement. What is the probability of choosing 5 hearts in a row?
• Researchers
  Researchers ask 200 families whether or not they were the homeowner and how many cars they had. Their response was homeowner: 14 no car or one car, two or more cars 86, not homeowner: 38 no car or one car, two or more cars 62. What percent of the families