Probability, Counting & Combinatorics
In just a few days, the 2019 MATHCOUNTS State Competitions will begin, so let’s look back at some of the 2018 State Sprint round problems and solve them in preparation.
Over the weekend, the world’s most famous groundhog, Punxsutawney Phil, enjoyed his favorite day of the year – Groundhog Day! According to historical data on Stormfax® Weather Almanac’s website (stormfax.com/ghogday), Phil has predicted a long winter 104 times, and an early spring 18 times from 1887 to 2018 (no data for nine of these years). That makes the odds of Phil seeing his shadow, thereby predicting six more weeks of winter for this year, 52 to 9. Based on this information, what is the percent probability of Phil seeing his shadow?
Chapter competitions officially start this week! Are you ready to compete? Perhaps solving these Sprint Round problems from the 2018 Chapter Competition will help you warm up.
By switching two of the digits of the number 123,456, Rodrigo obtains a new number that is 1980 more than the original. What is the product of the two digits Rodrigo switched?
Dr. Martin Luther King, Jr., a civil rights activist whose passion and leadership led to great cultural changes in America, was born on January 15, 1929. On November 2, 1983, President Ronald Reagan signed a bill to create a federal holiday honoring Dr. King. The national holiday celebrating the birth of Dr. King was first observed on January 20, 1986. Every year since, Martin Luther King, Jr. Day has been observed on the third Monday of January.
What is the earliest date in January that Martin Luther King, Jr. Day can occur?
As 2018 comes to an end, let’s go back and solve some of our favorite problems of year.
Problem 237: Kendra starts at a positive integer k and counts up by 4s until she hits exactly 200. Mason starts at a positive integer m and counts up by 6s until he hits exactly 200. If it takes Kendra half as many steps to reach 200 as it takes Mason, what is the greatest possible value of k − m?
Sam, Pam, Ken and Jen are exchanging gifts for the holiday. Each name is written on a slip of paper and placed in a bowl. Sam is the first to randomly select a name from the bowl. What is the probability that Sam selects his own name? Express your answer as a common fraction.
Santiago baked gingerbread cookies in the shapes of children. He gave each gingerbread boy and girl eyes made from chocolate covered candies in one of four different colors, a nose made from a butterscotch chip, toffee chip or peanut butter chip, and a mouth made from a piece of red licorice. Santiago then gave each gingerbread boy and girl buttons made from gum drops in one of four different flavors. For the final detail, he gave each gingerbread boy a blue bow tie, and each gingerbread girl a red hair bow.
Over the summer millions of people around the world visit amusement parks. Previously heralded as the “World’s Largest Amusement Park,” Cedar Point Amusement Park in Sandusky, Ohio earned that title. At one time, Cedar Point had 72 rides, 16 of which are roller coasters. What percent of the Cedar Point’s rides are not roller coasters? Express your answer as a decimal to the nearest tenth.
Moo’s Ice Cream shop just opened for the summer season. Moo’s has five different flavors of ice cream – chocolate, vanilla, strawberry, bubble gum and mint chocolate, as well as, three topping choices – sprinkles, whipped cream and chocolate chips. If a small sundae comes with one scoop and one topping, a medium sundae comes with two scoops and one topping and a large sundae comes with three scoops and two toppings. If medium and large sundaes can come with scoops of the same or different flavors of ice cream, how many different ice cream sundae combinations can be ordered at Moo’s?