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Since the math is already laid out I'll try to provide some intuition. The issue is that knowing that at least one frog is male is different from knowing that any particular frog is male, and the former case carries less information and this effectively increases our chances over the latter situation.

Call the frogs left and right, and suppose we are told that the right frog is male. Then we have eliminated two possible events from the sample space: the event where both frogs are female and the event where the left frog is male and the right frog is female. Now the probability truly is one half and it doesn't matter which one we choose. The exact same argument is true if we learn that the left frog is male.

But if we are told only that at least one frog is male, which is what happens when we hear the croak, then we cannot eliminate the event that the left frog is male and the right frog is female. We can only eliminate the event that both are female, which makes the event that at least one is female more likely than the previous setting.

I think the reason why this is confusing is that we naturally think learning that at least one is male should make us disinclined to choose the pair of frogs. It is true that this information makes it less probable that at least one is female, but recognize also that there was a full three quarters chance of at least one female before we learned anything at all. It's the ambiguity of the information we receive which makes it so we should still prefer the two frogs over the one.

Since the math is already laid out I'll try to provide some intuition. The issue is that knowing that at least one frog is male is different from knowing that any particular frog is male. The former case carries less information and this effectively increases our chances over the latter situation.

Call the frogs left and right, and suppose we are told that the right frog is male. Then we have eliminated two possible events from the sample space: the event where both frogs are female and the event where the left frog is male and the right frog is female. Now the probability truly is one half and it doesn't matter which one we choose. The exact same argument is true if we learn that the left frog is male.

But if we are told only that at least one frog is male, which is what happens when we hear the croak, then we cannot eliminate the event that the left frog is male and the right frog is female. We can only eliminate the event that both are female, which makes the event that at least one is female more likely than the previous setting.

I think the reason why this is confusing is that we naturally think learning that at least one is male should make us disinclined to choose the pair of frogs. It is true that this information makes it less probable that at least one is female, but recognize also that there was a full three quarters chance of at least one female before we learned anything at all. It's the ambiguity of the information we receive which makes it so we should still prefer the two frogs over the one.

Since the math is already laid out I'll try to provide some intuition. The issue is that knowing that at least one frog is male is different from knowing that any particular frog is male, and the former case carries less information and this effectively increases our chances over the latter situation.

Call the frogs left and right, and suppose we are told that the right frog is male. Then we have eliminated two possible events from the sample space: the event where both frogs are female and the event where the left frog is male and the right frog is female. Now the probability truly is one half and it doesn't matter which one we choose. The exact same argument is true if we learn that the left frog is male.

But if we are told only that at least one frog is male, which is what happens when we hear the croak, then we cannot eliminate the event that the left frog is male and the right frog is female. We can only eliminate the event that both are female, which makes the event that at least one is female more likely than the previous setting.

I think the reason why this is confusing is that we naturally think learning that at least one is male should make us disinclined to choose the pair of frogs. It is true that this information makes it less probable that at least one is male, but recognize also that there was a full three quarters chance of at least one female before we learned anything at all. It's the ambiguity of the information which makes it so we should still prefer the two frogs over the one.

Since the math is already laid out I'll try to provide some intuition. The issue is that knowing that at least one frog is male is different from knowing that any particular frog is male, and the former case carries less information and this effectively increases our chances over the latter situation.

Call the frogs left and right, and suppose we are told that the right frog is male. Then we have eliminated two possible events from the sample space: the event where both frogs are female and the event where the left frog is male and the right frog is female. Now the probability truly is one half and it doesn't matter which one we choose. The exact same argument is true if we learn that the left frog is male.

But if we are told only that at least one frog is male, which is what happens when we hear the croak, then we cannot eliminate the event that the left frog is male and the right frog is female. We can only eliminate the event that both are female, which makes the event that at least one is female more likely than the previous setting.

I think the reason why this is confusing is that we naturally think learning that at least one is male should make us disinclined to choose the pair of frogs. It is true that this information makes it less probable that at least one is female, but recognize also that there was a full three quarters chance of at least one female before we learned anything at all. It's the ambiguity of the information we receive which makes it so we should still prefer the two frogs over the one.

Since the math is laid out in the video I'll try to provide some intuition. The issue is that knowing that at least one frog is male is different from knowing that any particular frog is male, and the former case carries less information which effectively increases our chances over the latter situation.

Call the frogs left and right, and suppose we are told that the right frog is male. Then we have eliminated two possible events from the sample space: the event where both frogs are female and the event where the left frog is male and the right frog is female. Now the probability truly is one half and it doesn't matter which one we choose. The exact same argument is true if we learn that the left frog is male.

But if we are told only that at least one frog is male, which is what happens when we hear the croak, then we cannot eliminate the event that the left frog is male and the right frog is female. We can only eliminate the event that both are female, which makes the event that at least one is female more likely than the previous setting.

I think the reason why this is confusing is that we intuitively think that learning that at least one is male should make us disinclined to choose the pair of frogs. It is true that this information makes it less probable that at least one is male, but recognize also that there was a full three quarters chance of at least one female before we learned anything at all. It's the ambiguity of the information which makes it so we should still prefer the two frogs over the one.

Since the math is already laid out I'll try to provide some intuition. The issue is that knowing that at least one frog is male is different from knowing that any particular frog is male, and the former case carries less information and this effectively increases our chances over the latter situation.

Call the frogs left and right, and suppose we are told that the right frog is male. Then we have eliminated two possible events from the sample space: the event where both frogs are female and the event where the left frog is male and the right frog is female. Now the probability truly is one half and it doesn't matter which one we choose. The exact same argument is true if we learn that the left frog is male.

But if we are told only that at least one frog is male, which is what happens when we hear the croak, then we cannot eliminate the event that the left frog is male and the right frog is female. We can only eliminate the event that both are female, which makes the event that at least one is female more likely than the previous setting.

I think the reason why this is confusing is that we naturally think learning that at least one is male should make us disinclined to choose the pair of frogs. It is true that this information makes it less probable that at least one is male, but recognize also that there was a full three quarters chance of at least one female before we learned anything at all. It's the ambiguity of the information which makes it so we should still prefer the two frogs over the one.