A card from a pack of 52 cards is lost. From the remaining 51 cards, n cards are drawn and are found to be spades. If the probability of the lost card being a spade is 5011, then n is equal to:
Get the complete, step-by-step math solution for: "A card from a pack of 52 cards is lost. From the remaining 51 cards, n cards are drawn and are found to be spades. If the probability of the lost card...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Define Events and Probabilities
Let LS be the event that the lost card is a spade, and LNS be the event that the lost card is not a spade. A standard deck has 13 spades and 39 non-spades. We calculate the initial probabilities for these events.
Step 2: Calculate Conditional Probabilities of Drawing Spades
Let E be the event that n cards drawn from the remaining 51 cards are all spades. If the lost card was a spade (LS), there are 12 spades left in the 51 cards. If the lost card was not a spade (LNS), there are 13 spades left in the 51 cards. We use combinations to find the probability of drawing n spades in each case.
Step 3: Apply Bayes' Theorem
We are given the probability of the lost card being a spade, given that n cards drawn are spades, which is P(LS∣E)=5011. We use Bayes' theorem to relate this to the probabilities we calculated.
Step 4: Substitute and Simplify
Substitute the probabilities into Bayes' theorem. The term 4(n51)1 cancels out from the numerator and denominator, simplifying the expression.
Step 5: Expand Combinations and Solve for n
We expand the combination terms using the formula (rk)=rk!!(k−r)!. After simplification, we get an equation in terms of n. We then solve this algebraic equation to find the value of n.