A game begins with two cards being dealt from a standard deck of 2 cards. To win this game, the next card must be the same as either of these first two cards, or fall between them. If the first two cards are a 4 and a 9, what is the probability of winning the game?
you need a 4,5,6,7,8,9
There are 20 cards(above) which are in the deck to win, of a deck of 50 cards.
Pr(win)=20/50
To calculate the probability of winning the game, we need to determine the number of favorable outcomes and the total number of possible outcomes.
The favorable outcomes in this case are the cards that can be drawn next, which are any card that is a 4 or a 9, or any card that falls between 4 and 9.
For the first part, we need to find the number of cards that are 4 or 9 in the deck. Since there are four suits in a standard deck, and each suit has one 4 and one 9, we have a total of 2 favorable outcomes for the first card.
For the second part, we need to find the number of cards that fall between 4 and 9. These would be the cards 5, 6, 7, and 8. Again, considering four suits, we have a total of 4 * 4 = 16 favorable outcomes for the second card.
Now, let's find the total number of possible outcomes. We already have the first two cards drawn, so we only need to consider the third card. Since there are two cards already drawn, we have 52 - 2 = 50 cards left in the deck.
Therefore, the total number of possible outcomes is 50.
The probability of winning the game is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. In this case, the number of favorable outcomes is 2 + 16 = 18.
Thus, the probability of winning the game is 18/50, which simplifies to 9/25 or 0.36, or 36%.