For the experiment of drawing a single card from a standard 52 -card deck, find (a) the probability of the following event, and (b) the odds in favor of the following event. Neither a spade nor an ace

To solve the problem, we need to find the probability and the odds in favor of drawing a card that is neither a spade nor an ace from a standard 52-card deck. ### (a) Probability 1. **Total number of cards in the deck**: 52 2. **Number of spades in the deck**: 13 (one for each rank) 3. **Number of aces in the deck**: 4 (one for each suit: spades, hearts, diamonds, clubs) However, one of the aces is also a spade (the Ace of Spades). Therefore, we need to be careful not to double-count it. 4. **Total number of cards that are either a spade or an ace**: - Number of spades: 13 - Number of aces: 4 - Subtract the Ace of Spades (which is counted in both categories): 1 So, the total number of cards that are either a spade or an ace is: \[ 13 + 4 - 1 = 16 \] 5. **Number of cards that are neither a spade nor an ace**: \[ 52 - 16 = 36 \] 6. **Probability of drawing a card that is neither a spade nor an ace**: \[ P(\text{neither a spade nor an ace}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{36}{52} = \frac{9}{13} \] ### (b) Odds in favor The odds in favor of an event are calculated as the ratio of the number of favorable outcomes to the number of unfavorable outcomes. 1. **Number of favorable outcomes** (neither a spade nor an ace): 36 2. **Number of unfavorable outcomes** (either a spade or an ace): 16 3. **Odds in favor of drawing a card that is neither a spade nor an ace**: \[ \text{Odds in favor} = \frac{\text{Number of favorable outcomes}}{\text{Number of unfavorable outcomes}} = \frac{36}{16} = \frac{9}{4} \] ### Summary - **Probability of drawing a card that is neither a spade nor an ace**: \(\frac{9}{13}\) - **Odds in favor of drawing a card that is neither a spade nor an ace**: \(\frac{9}{4}\)