Anna, Brad, and Pennies
"Come over to our store for a chance of winning with your amazing luck!" Brad said over through the speakers at a Math Carnival Fair.
Not long later, a girl came over to Brad's store to see what the store has in store for her!
"Hello!" Brad said to the young girl. "What's your name?"
"Anna." The girl replied. "So what is this game about?" she questioned.
"Oh, it's simple." Brad said to Anna. "All you need to do is give me a penny for every throw of 6 coins, and if all of them come up heads, you win 60 pennies back!"
"Alright then, let me have a go!" she said confidently. Anna gave 1 penny to Brad and had a throw of the coins. 5 tails, 1 head.
"Want to try again?" Brad said.
"Okay then!" Anna gave another penny to Brad and throw the coins even harder onto the table in the store. 2 tails, 4 heads.
Before Brad could ask Anna for another penny, Anna immediately gave her 3rd penny to Brad, picked up the 6 coins and threw the coins very, very hard onto the table such that all the 6 coins rolled off the table! Brad went off crawling under the table, trying to pick up the rolling coins, but did not notice that Anna secretly took 1 of the 6 coins on the floor and kept it in her pocket.
After a minute, Brad told Anna that he only found 5 of the 6 coins.
"No worries! Since there are only 5 coins now, you can make the grand prize 50 pennies instead!" Anna said to Brad, trying to comfort him. "The odds of me winning is still the same, as the ratio of the number of coins to the grand prize is still 1:10!"
"That sounds like a brilliant idea!" Brad said to Anna. So, want to have a roll of the coins?"
"Sure!"
Hence, Anna keeps giving a penny to Brad, and having a throw of the remaining 5 coins.
After a lot of throws have been made, who will lost out at the end, Anna or Brad?
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1 solution
The odds that all 6 coins will turn up heads is 1/2^6, which is 1/64! That means on average, every 64 pennies Anna give to Brad, only 60 pennies will be returned to her. After a lot of throws, Anna will obviously lose out.
But now the situation has changed. The odds that all the remaining 5 coins will turn up heads is 1/2^5, which is 1/32! That means on average, every 32 pennies Anna give to Brad, 50 pennies will be returned to her! After a lot of throws, Brad will obviously lose out.
Hence, the person who loses out at the end is Brad .